Introduction

So, we've found some new results about infradistributions, and accordingly, this post will be a miscellaneous grab-bag of results from many different areas. Again, this is a joint Vanessa/Diffractor effort. It requires having read "Basic Inframeasure Theory" as a mandatory prerequisite, and due to a bit less editing work this time around, it's advised to pester me for a walkthrough of this one, illusion of transparency is probably worse on this post than the previous two. Apart from the 8 proof sections (1, 2, 3, 4, 5, 6, 7, 8), included in this post is:

Section 1: We cover generalizations of the background mathematical setting of inframeasure theory to cover discontinuous functions, taking the expectation of functions outside of $[0,1]$, and defining infradistributions over non-compact spaces. We'll also cover what "support" means for infradistributions, and why it becomes important when we go outside of compact spaces.

Section 2: We'll introduce various convenient properties that infradistributions might fulfill, from weakest to strongest, what they mean for the expectation functionals as well as the set of minimal points, and what they're used for.

Section 3: We present several operations to form new infradistributions from old ones, and talk about what they mean, both on the set side of LF-duality as well as the expectation functional side. We go more in-depth on things like semidirect product which were casually mentioned at the end of the last post, as well as some novel ones like pullback.

Section 4: We briefly introduce the basics of optimistic Knightian uncertainty, which we'll call ultradistributions, as opposed to pessimistic Knightian uncertainty/infradistributions. It's covered in only cursory depth, because they appear so closely parallel to our existing framework that if ultradistributions are ever required for something, it would be very easy to adapt the requisite results from the theory of infradistributions.

Section 5: We present our first major theorem, the analogue of the disintegration theorem from classical probability theory. It is a highly useful tool which gives the infra-version of conditional probability. Its use is more restricted than the probability theory analogue, but it lets you do things like derive the analogue of Bayes' rule when you have Knightian uncertainty as to the prior probability distribution over hypotheses. Also we have an analogue of Markov's inequality.

Section 6: This section addresses notions of distance between infradistributions. In particular, we have a notion of distance for infradistributions which, in the standard probabilistic case, reduces to the KR-metric we know and love. We characterize this metric and the space of infradistributions equipped with this metric in many different ways, including finding strongly equivalent versions of it for functionals and their associated sets (by LF-duality), locating necessary-and-sufficient conditions for a set of infradistributions to be compact according to this metric, showing that the topology it induces is a close cousin of the Vietoris topology, and using this topology to elegantly characterize infrakernels (plus one additional condition) as just bounded continuous functions.

Section 7: The basics of Infra-Markov processes. This is just giving a few definitions, and will mostly be followed up on in a future post about infra-POMDP's. However, we also have a far-reaching generalization of the classic result where Markov chains (fulfilling some conditions) have a stationary distribution which can be found by iterating. Infra-Markov chains (fulfilling some conditions) have a stationary infradistribution which can be found by iterating.

Section 8: When an infradistribution is over finitely many events, we can define entropy for it. We define the Renyi entropy for $\alpha \in (1,\infty )$, use that to derive a few different formulations of the Shannon entropy, cover what entropy intuitively means for crisp infradistributions, and show a few basic properties of entropy.

Section 9: Teasers for additional work.

EDIT: In the comments, we work out what the analogue of KL-divergence is for crisp infradistributions, and show it has the basic properties you'd expect from KL-distance.

Section 0: Notation Glossary

$X$: A Polish space.

$C(X,[0,1\left]\right)$: The space of continuous functions $X\to [0,1]$.

$C_B\left(X\right)$: the Banach space of continuous bounded functions $X\to R$, equipped with the sup-norm, ${sup}_{x\in X}|f\left(x\right)|$.

$f,f^{\prime },f^{\ast }$: Functions. Typically of type signature $C_B\left(X\right)$.

$f_{\downarrow B}$: The restriction of a function f to some set $B$.

$a,c,p$: Constants. Generally, $a,c$ are in $R$ or $R^{\ge 0}$, while $p\in [0,1]$ and represents a probability you use to mix two things together.

$d(f,f^{\prime }),d(m,m^{\prime }),d(x,x^{\prime })$: Various distance metrics. There are some others that we try to describe with subscripts, but if you're looking at distances between functions, we're using the sup-norm distance metric $d(f,f^{\prime }):={sup}_x|f\left(x\right)-f^{\prime }\left(x\right)|$. If you're lookin at distances between measures, we're using the KR-metric, $d(m,m^{\prime }):={sup}_{f\in C_{1-Lip}(X,[-1,1\left]\right)}|m\left(f\right)-m^{\prime }\left(f\right)|$ (ie, if you feed in a 1-Lipschitz function bounded to be near 0, what's the biggest difference in the expectation values you can muster), and if you're looking at distances between points $x,x^{\prime }$, we're just equipping the space $X$ with some distance metric.

$||f||$: The quantity $d(f,0)$, aka ${sup}_x|f\left(x\right)|$.

$Li\left(f\right)$: If $f$ is a Lipschitz function, it's the Lipschitz constant of $f$.

$\mu \left(f\right),m\left(f\right),m\left(A\right)$: The expectation of a function w.r.t. some probability distribution, or measure, or the measure on some set $A$. Like, $m\left(f\right):={\int }_{x\in X}f\left(x\right)dm$.

$h$: An infradistribution functional, of type signature $C_B\left(X\right)\to R$, fulfilling some properties like monotonicity and concavity.

$H$: An infradistribution set, a closed convex upper-complete set of a-measures fulfilling a few other conditions.

$\square X$: The space of infradistributions over the space $X$.

$\Delta X$: The space of probability distributions over the space $X$.

$(m,b)$: An a-measure. $m$ is a measure, $b$ is a number that's 0 or higher.

$M^a\left(X\right)$: The set of a-measures over the space $X$.

$B^{uc}$: the upper completion of a set of a-measures $B$, made by the set $B+\left\{\right(0,b)|b\ge 0\}$, where + is Minkowski sum.

$H^{min}$: the minimal points of the infradistribution $H$, those points $(m,b)$ where the $b$ value can't get any lower without the point traveling outside $H$.

$(\lambda \mu ,b)$: An alternate representation of an a-measure, where we can break the measure down into a probability distribution $\mu$ and a scaling term $\lambda$. Talk of $\lambda$ and $b$ for a-measures is referring to these quantities of "how much is the amount of measure present" and "how much of the utility is present".

$E_H\left(f\right)$: The expectation of a function according to an infradistribution set, defined as $E_H\left(f\right)={inf}_{(m,b)\in H}m\left(f\right)+b$

${\lambda }_h^{\odot },{\lambda }_K^{\odot }$: Used to denote the Lispchitz constant of some infradistribution or infrakernel. In other words, ${\lambda }_h^{\odot }={sup}_{f,f^{\prime }}\frac{|h\left(f\right)-h\left(f^{\prime }\right)|}{d(f,f^{\prime })}$ The subscript helps denote which infradistribution or infrakernel it came from, if there's ambiguity about that.

$L,g$: In the context of updates, $L:C(X,[0,1\left]\right)$ and is a likeliehood function saying how likely various points are to produce some observation, while $g:C_B\left(X\right)$ is an "off-history utility function" used to define the update.

$f{★}^{L}g$: An abbreviation for the sorts of functions we take the expectation of when we define the update, defined as:
$\left(f{★}^{L}g\right)\left(x\right):=L\left(x\right)f\left(x\right)+(1-L(x\left)\right)g\left(x\right)$
Where $L:C(X,[0,1\left]\right)$ and $g,f:C_B\left(X\right)$.

$h{|}^{g}L:$ An infradistribution, updated on off-event utility $g$ and likelihood function $L$, defined as:
$\left(h{|}^{g}L\right)\left(f\right):=\frac{h\left(f{★}^{L}g\right)-h\left(0{★}^{L}g\right)}{h\left(1{★}^{L}g\right)-h\left(0{★}^{L}g\right)}$

$P_h^g\left(L\right)$: The quantity $h\left(1{★}^{L}g\right)-h\left(0{★}^{L}g\right)$.

$E_{\zeta }{h}_i$: A mixture of infradistributions according to some probability distribution $\zeta \in \Delta N$, defined as $\left(E_{\zeta }{h}_i\right)\left(f\right):=E_{\zeta }\left(h_i\right(f\left)\right)$

$g_{\ast }\left(h\right)$: Given some continuous function $g:X\to Y$, and infradistribution $h:\square X$, this is the pushforward infradistribution of type $\square Y$, defined by
$g_{\ast }\left(h\right)\left(f\right):=h(f\circ g)$

$inf(h_1,h_2)$: The infinimum of two infradistributions according to information ordering, defined by:
$inf(h_1,h_2)\left(f\right):=inf\left(h_1\right(f),h_2(f\left)\right)$

$sup(h_1,h_2)$: The supremum of two infradistributions according to information ordering, defined by
$sup(h_1,h_2)\left(f\right):={sup}_{p,f_1,f_2:p{f}_1+(1-p)f_2\le f}p{h}_1\left(f_1\right)+(1-p)h_2\left(f_2\right)$

$K$: Usually used for an infrakernel, a function $X\to \square Y$ with the shared Lipschitz-constant property, pointwise convergence property, and compact-shared CAS property. Described in the post.

$\stackrel{ik}{\to }$: $X\stackrel{ik}{\to }Y$ is the type of infrakernels $X\to \square Y$.

$\lambda x.(\lambda y.f(x,y\left)\right)$: Lambda notation for functions. This would be "the function that maps $x$ to (the function that maps $y$ to $f(x,y)$)".

$⋉$: The semidirect product, $h⋉K$, where $h\in \square X$ and $K:X\stackrel{ik}{\to }Y$, is an infradistribution $\square (X\times Y)$ defined by:
$(h⋉K)\left(f\right):=h(\lambda x.K(x\left)\right(\lambda y.f(x,y)\left)\right)$

$\times$: The direct product, $h_1\times h_2$, where $h_1\in \square X$, $h_2\in \square Y$, is an infradistribution $\square (X\times Y)$ defined by:
$(h_1\times h_2)\left(f\right):=h_1(\lambda x.h_2(\lambda y.f(x,y)\left)\right)$

$p{r}_X$: The projection map from some space $X\times Y$ to $X$. The thing in the subscript tells you what you're projecting to.

$\oplus$: The coproduct, $h_1\oplus h_2$, where $h_1\in \square X$, $h_2\in \square Y$, is an infradistribution $\square (X+Y)$ defined by:
$h_1\oplus h_2:=inf\left(i_{1\ast }\right(h_1),i_{2\ast }(h_2\left)\right)$
Where $i_1$ and $i_2$ are the injection maps.

$K_{\ast }$: the infrakernel pushforward, $K_{\ast }\left(h\right)$, where $h\in \square X$ and $K:X\stackrel{ik}{\to }Y$, is an infradistribution $\square Y$ defined by:
$K_{\ast }\left(h\right)\left(f\right):=h(\lambda x.K(x\left)\right(f\left)\right)$

$k_{\ast }$ is the same thing, except when your infrakernel is just a Feller-continuous function $X\to \Delta Y$.

$g^{\ast }$: the pullback. If $g$ is a continuous proper map $X\to Y$, and $h$ is an infradistribution on $Y$ supported on $g\left(X\right)$, then $g^{\ast }\left(h\right):\square X$, and is defined by:
$g^{\ast }\left(h\right)\left(f\right):={sup}_{f^{\prime }:f^{\prime }\circ g\le f}h\left(f^{\prime }\right)$

$\ast$: The free product of infradistributions. Given two infradistributions $h_1\in \square X$ and $h_2\in \square Y$, then $h_1\ast h_2:\square (X\times Y)$ is the infradistribution given by:
$(h_1\ast h_2):=sup\left(p{r}_X^{\ast }\right(h_1),p{r}_Y^{\ast }(h_2\left)\right)$

$x_{n:m}$ This subscript notation generalizes to other things. When you've got a sequence of thingies (let's say points $x_n$ from spaces $X_n$), then $x_{n:m}$ refers to a point from ${\prod }_{i=n}^{i=m}{X}_i$. $x_{0:m}$ gets abbreviated as $x_{:m}$, and $x_{m:\infty }$ refers to a point from ${\prod }_{i=m}^{\infty }{X}_i$. In general, this notation applies to any time when you're composing or sticking together a bunch of things indexed by numbers and you need to keep track of where you start and where you end.

$\text{support}\left(f\right)$: The support of a function $f:X\to R^{\ge 0}$, the set $\left\{x|f\right(x)>0\}$.

$\overline{B}$: The closure of a set $B$. The overline is closure of a set.

$c.h\left(B\right)$: The convex hull of a set $B$.

$C_{ϵ}^X$: A compact $ϵ$-almost-support. Generally, the superscript tells you something about what space this is a subset of, and the subscript tells you what level of almost-support it is. Compact almost-supports are defined later.

$K\left(X\right)$: the space of nonempty compact subsets of the space $X$, equipped with the Vietoris topology.

$dh(f;f^{\prime })$: The Gateux derivative of infradistribution $h$ at $f$ in the direction of the function $f^{\prime }$, defined as ${lim}_{ϵ\to 0}\frac{h(f+ϵf^{\prime })-h\left(f\right)}{ϵ}$.

$H_{\alpha }\left(h\right)$: the $\alpha \in (1,\infty )$-Renyi entropy of infradistribution $h$.

$H\left(h\right)$: When no subscript is specified, this refers to the $\alpha \to 1$ limit of $H_{\alpha }\left(h\right)$, ie, the Shannon entropy of the infradistribution $h$ (reduces to standard Shannon entropy for probability distributions)

$H(\mu ,\nu )$: The cross-entropy of two probability distributions, the quantity $-{\sum }_i{\mu }_i\ln \left({\nu }_i\right)$.

Section 1: Generalizations

There's three notable variants on the mathematical framework found so far. Our old mathematical framework was viewing an infradistribution as a set of sa-measures over a compact metric space $X$, fulfilling some special properties. Or, by LF-duality, an infradistribution is a concave monotone functional of type signature $C(X,[0,1\left]\right)\to [0,1]$ (it intakes a continuous function $X\to [0,1]$ and spits out an expectation value).

Our three directions of generalization are: Permitting the use of the expectations of non-continuous functions, permitting expectations to be taken of any bounded continuous function instead of having the $[0,1]$ restriction, and letting our space $X$ be something more general than a compact metric space. The first path won't be taken, but the latter two will. Going to bounded continuous functions in general makes the theory of distances between infradistributions much tidier than it would otherwise be, as well as making the supremum operation and free product behave quite nicely. And letting our space $X$ be something more general than a compact metric space lets us have infradistributions over spaces like the natural numbers and do updates properly, and leads to an improved understanding of which conditions are necessary for an infradistribution to be an infradistribution.

Generalization to Measurable Functions

So, it may seem odd that we can only take the expectation of continuous functions. Probability distributions can take expectations of measurable functions, right? Shouldn't we be able to do the same for infradistributions? Being able to take expectations of measurable functions instead of just continuous functions would let you do updates on arbitrary measurable sets, instead of having to stick to clopen sets or fuzzy sets with continuous indicator functions.

On the concave functional side of LF-duality, we can do this by swapping out $C_B\left(X\right)$, the space of continuous bounded functions $X\to R$, for $B\left(X\right)$, the space of bounded measurable functions $X\to R$, and equipping it with the metric of uniform convergence (ie, $d(f,f^{\prime })={sup}_x|f\left(x\right)-f^{\prime }\left(x\right)|$)

To preserve LF-duality, on the set side, we'd have to go to the space of finitely additive bounded measures on $X$. Ie, if $A_i$ are all disjoint subsets of our space $X$, instead of $\mu \left({\bigcup }_{i=0}^{\infty }{A}_i\right)={\sum }_{i=0}^{\infty }\mu \left(A_i\right)$ (countably additive), we can only do this for finitely many disjoint sets. Most of measure theory looks at countably additive measures instead of finitely additive ones, so my hunch is that it'd introduce several subtle measure-theoretic issues throughout, instead of just being able to cleanly apply basic textbook results, and make our space of measures "too big". But it can probably be done if we really need discontinuous functions for something, we'd just have to go through our proofs with a fine-toothed comb. Finitely additive measures instead of countably additive ones are fairly odd. For instance, a nonprincipal ultrafilter over $N$ (1 if the set is in the ultrafilter, 0 if it isn't) is a finitely additive measure that isn't countably additive. So we'd have to consider strange creatures like that.

Generalization to Functions Outside [0,1]

Having to deal with functions in $[0,1]$ works nicely for utility functions, but maybe we want more generality. So, instead of having our infradistributions be defined over $C(X,[0,1\left]\right)$, (continuous functions $X\to [0,1]$) we could have them be defined over $C_B(X,R^{\ge 0})$ (bounded continuous functions that are $\ge 0$), or even $C_B\left(X\right)$ (bounded continuous functions) in full generality.

In fact, to get the supremum and free product and metrics on the space of infradistributions, to behave nicely, we have to switch to our infradistributions having type signature $C_B\left(X\right)\to R$. Infradistributions of type signature $C(X,[0,1\left]\right)\to [0,1]$ have a... more strained relationship with these concepts, so in the interests of mathematical tidiness, we'll be we working with infradistributions of type signature $C_B\left(X\right)\to R$ from here on out.

There are three advantages and three disadvantages to making this switch. The disadvantages are: 

1: Infradistributions of type signature $C(X,[0,1\left]\right)\to [0,1]$ cut down on a bunch of extra data that an infradistribution of type $C_B\left(X\right)\to R$ specifies. If your utility function is in $[0,1]$, for example, the ability to take expectations of functions outside of $[0,1]$ is rather pointless.

2: The particular form of normalization we use is most naturally tailored to infradistributions of type $C(X,[0,1\left]\right)\to [0,1]$, because it's basically "send the worst-case function, the constant-0 function, to a value of 0, and the best-case function, the constant-1 function, to a value of 1", and the scale-and-shift of inframeasures (fulfill everything but normalization) to make an infradistribution is perfectly analogous to rescaling a bounded utility function to lie in [0,1]. However, when you've got type signature $C_B\left(X\right)\to R$, this becomes more arbitrary. Obviously, 0 should be mapped to 0. But mapping 1 to 1? Why not scale-and-shift to map -1 to -1, or make it so the slope of the functional $h$ at 0 in the direction of increasing the constant is 1? We'll stick with the old "0 to 0, 1 to 1" normalization, but maybe another one will be developed that has better properties. I'll let you know if one shows up.

3: There's a concept called isotonic infradistributions, which are closely tied to proper scoring rules, and the concept only behaves well for infradistributions of type signature $C(X,[0,1\left]\right)\to [0,1]$. However, these will not be discussed in this post.

What about the advantages?

1: We can evaluate the expectation of any bounded function, instead of just functions bounded in $[0,1]$. Although that restriction works nicely for utility functions, it's fairly restrictive in general. If infradistributions are to serve as a strictly superior generalization of probability theory... well, probability theory often involves expectations of functions not bounded in $[0,1]$, so we should be able to accommodate such things.

2: As we'll see later, the set form of these sorts of infradistributions only involves a-measures (a pair of a measure and a nonnegative b term, without any mucking around with signed measures), which is very convenient for proving results. Also, the notion of "upper completion" for the set form of an infradistribution becomes much more manageable, and it's trivial to find minimal points. And finally, the KR-distance doesn't actually metrize the weak topology on signed measures, but it does metrize the weak topology on measures, so we don't have to worry about rigorously cleaning up any fiddly measure-theoretic issues regarding the use of signed measures.

3: But by far the most important reason to switch to infradistributions of type signature $C_B\left(X\right)\to R$ is that it just interacts much better with various operations. The supremum behaves much better, the free product behaves much better, the space of infradistributions can be made into a Polish space, the additional compactness condition we must add when we go to Polish spaces has a natural interpretation, you get a close match between infrakernels and bounded continuous functions... This type signature is unsurpassed at equipping infradistributions with nice properties, and I eventually gave up after a month of trying to make $C(X,[0,1\left]\right)\to [0,1]$ infradistributions behave nicely w.r.t. those things. This isn't a rushed decision to switch to this type signature, I was forced into it.

So, from now on, when we say "infradistribution", we're talking about the type signature $C_B\left(X\right)\to R$, fulfilling some additional conditions, unless specifically noted otherwise.

Generalization to Polish Spaces

This is another generalization that we'll be adopting from this point onwards unless explicitly noted. Our old definition of what a permissible space to have infradistributions over was: $X$ is a compact metric space. But we can generalize this significantly, to Polish spaces (which have compact metric spaces as special cases)

Broadening our notion of what spaces are permissible to have infradistributions over does come at the cost of introducing some extra complexity in the proofs and results, but we'll specifically note when we're making extra assumptions on the relevant space.

The concept of Polish spaces is used throughout the rest of this post, so this section is very important to read so you have some degree of fluency in what a Polish space is.

Previously, we were limited to compact separable complete metric spaces to define infradistributions over. (this is redundant, because compact and metric imply the other two properties). Polish spaces are just separable complete metric spaces.

Technically, this is slightly wrong, because Polish spaces only have a topology specified, not a metric. So, you can think of them more accurately as being produced by starting with a separable complete metric space, and forgetting about the metric, just leaving the topology alone. But these details are inessential most of the time.

Some exposition of what these conditions mean is in order.

"Metric" just means it has a notion of distance. "Complete" means that all Cauchy sequences in the metric (sequences which ought to converge) have their limit point already present in the space. For example, $Q$ isn't complete under the usual metric, but $R$ is. Taking limits shouldn't lead you outside the relevant space.

And finally, "separable" means that there's a countable dense subset. Ie, there's countably many points where, given any point in the space, you can find something from your countable list arbitrarily close to it. As a concrete example, $Q$ shows that $R$ fulfills this property. There's countably many fractions, and, given any real number, you can find a fraction arbitrarily close to it. In particular, this implies that a Polish space can't be "too big". It must have the cardinality of the continuum or less, because everything can be written as a sequence of countably many points from a countable set. You can't necessarily express any point in it with finite data, but you can express arbitrarily close approximations to any point in it with finite data.

So, that's what a Polish space is. It's a space which can be equipped with a notion of distance where limits don't lead you outside of the space, and there's a countable dense subset. That last one is critical because, if it's missing, you can give up all hope of being able to work with such a space on a computer. You may not be able to deal with arbitrary real numbers on a computer, but the presence of a countable dense subset (rationals) means that you can round a real number off a tiny bit to a rational number, and do operations on those. But if a space isn't separable, then there must be points in it which can't be arbitrarily closely approximated with finite data you can work with on a computer.

Concrete examples of Polish spaces: All the previously mentioned spaces in the last post, $N$, $R$, any closed or open subset of a Polish  space, or countable product of Polish spaces, the space $C\left(X\right)$ of continuous functions $X\to R$ when $X$ is compact (more generally, any separable Banach space is Polish), and the space of probability distributions over $X$ when $X$ is Polish (equipped with the weak topology). So, if you want infradistributions over the natural numbers, or over the real line, or over probability distributions on a Polish space, or continuous functions on a compact set, you can have them. This is a fairly large leap in generality.

So... how do we accommodate Polish spaces in our framework? We've just talked about what they are. Well, because $X$ isn't compact any more, our function space for the functional side of LF-duality has to shift from $C\left(X\right)$ (the space of continuous functions $X\to R$, because continuous functions on a compact space are bounded automatically) to $C_B\left(X\right)$ (the space of bounded continuous functions $X\to R$).

There's two problems we run into when we make this shift. Compactness is an incredibly handy property, and we're no longer able to exploit our old compactness lemma, which showed up in a bunch of proofs in "Basic Inframeasure Theory". Second, due to not being in a compact space, we lose the ability to casually identify continuous functions and uniformly continuous functions, they can be different now, which was also used a lot in proofs.

To get around these, we'll have to eventually impose a condition on infradistributions which lets us work in the compact setting, modulo a little bit of error (the compact almost-support property)

Updating the Right Way:

An extremely convenient feature we have is that open subsets of a Polish metric space are Polish, such as the open interval $(0,1)$ (though you do need to change your metric, the standard metric inherited from $R$ says that space isn't complete, it's missing the points 0 and 1.) This lets us deal with updating in full generality, which we'll now go into.

As a recap, the way we dealt with updating on a likelihood function (with $L:C(X,[0,1\left]\right)$ as the likelihood function, $g:C_B\left(X\right)$ as the function giving you your off-event utility, and the starting infradistribution is $h$) was defining the updated infradistribution by:

$\left(h{|}^{g}L\right)\left(f\right):=\frac{h\left(f{★}^{L}g\right)-h\left(0{★}^{L}g\right)}{h\left(1{★}^{L}g\right)-h\left(0{★}^{L}g\right)}$

Where $f{★}^{L}g:C_B\left(X\right)$ is defined by:

$\left(f{★}^{L}g\right)\left(x\right):=L\left(x\right)f\left(x\right)+(1-L(x\left)\right)g\left(x\right)$

Ie, you blend the functions together with $L$. When $L$ is near 1, we're in the region which plausibly could have occurred, and we use $f$. When $L$ is near 0, we're in the region which has been ruled out, and we use $g$, as that's our off-event utility function. For infradistributions, all the functions you're evaluating have to be continuous, so in particular, $f{★}^{L}g$ must be continuous to look at what its expectation is.

Now, originally, from the last post, $h{|}^{g}L$ was an infradistribution over the space $\overline{\text{support}\left(L\right)}$ (the region where $L\left(x\right)>0$, but take the closure), and f was a continuous bounded function from that space to $R$. Our post-update space had to be closed because we could only have infradistributions over compact spaces back then, and closed subsets of compact sets are compact, which is why we had to take the closure. But, actually, the support is an open set! And if we're working over Polish spaces, that's a-ok, we don't need to add anything special or clean it up, because an open subset of a Polish space is Polish.

As further evidence for not actually needing to take the closure in updating, we technically only need $f$ to be continuous on the support of $L$ for $f{★}^{L}g$ to be continuous. Like, $f$ could be continuous on the open set "support of $L$", but start having discontinuities at the "edges" of the set where $L$ is 0. But then the bad behavior of $f$ on the zero-likelihood edges would get crunched down when you multiply the function by $L$, so then $Lf$ would still be continuous, and thus $Lf+(1-L)g=f{★}^{L}g$ would be continuous and bounded and you could evaluate its value as usual with no issues.

To metrize the support of a likelihood function, just restrict the original distance metric on $X$ to be at most 1 and define your new metric $\left(d|L\right)(x,y):{\left(\text{support}\right(L\left)\right)}^2\to R^{\ge 0}$ as:

$\left(d|L\right)(x,y):=d(x,y)\cdot inf(\frac{1}{L\left(x\right)},\frac{1}{L\left(y\right)})+|\frac{1}{L\left(x\right)}-\frac{1}{L\left(y\right)}|$

Proposition 1: For a continuous function $L:X\to [0,1]$, the metric $d|L$ completely metrizes the set $\left\{x|f\right(x)>0\}$ equipped with the subspace topology.

Why do we care about such fiddly technicalities? Does it affect anything? Well, one of our upcoming results, the Infra-Disintegration Theorem, naturally produces some of those  functions that can be discontinuous on the zero-probability "edges" of the fuzzy set you're updating on, because conditional probability can vary wildly on low or zero-probability events. So now that we're no longer restricted to compact spaces, we can properly address updates like that.

Supports and Almost Supports

The largest change we make when we go to Polish spaces is that there's a prominent extra condition we must add to fairly call something an "infradistribution", because things get extremely messy if we don't. First, some exposition on the analogous property for ordinary probability distributions. If you have a probability distribution on $R$, it can't be a uniform distribution. It must tail off as you head towards infinity in either direction. In a more general form, any probability distribution on a Polish space has the property that there's a sequence of compact sets that more and more of the probability mass lies within, limiting to all of it. We'd want an analogous property for our infradistributions, as "compact set that accounts for arbitrarily-much-but-not-all of the infradistribution" would be perfect for replacing the compactness arguments that we used to use all over the place.

To properly formulate this, we must define a support. The notation $f_{\downarrow B}$ is the restriction of a function to a set $B$.

Definition 1: Support
A closed set $B\subseteq X$ is a support for a functional $h:C_B\left(X\right)\to R$ iff
$\forall f,f^{\prime }\in C_B\left(X\right):f_{\downarrow B}=f_{\downarrow B}^{\prime }\to h\left(f\right)=h\left(f^{\prime }\right)$

So, a support is just "if functions disagree outside of it, it doesn't matter for assessing their value". Note that we're speaking of a support instead of the support, but as we'll see a bit later, we just need to assume one additional condition to get that infradistributions have a unique closed set which could be fairly called the support. 

Now, the compactness condition on probability distributions we discussed earlier required that arbitrarily large fractions of the  probability mass can be accounted for a compact set, not that all of the probability mass be accounted for by a compact set. Accordingly, we define:

Definition 2: $ϵ$-Almost Support
A closed set $B\subseteq X$ is an $ϵ$-almost support for a functional $h:C_B\left(X\right)\to R$ iff
$\forall f,f^{\prime }\in C_B\left(X\right):f_{\downarrow B}=f_{\downarrow B}^{\prime }\to |h\left(f\right)-h\left(f^{\prime }\right)|\le ϵ\cdot d(f,f^{\prime })$

For this, our distance is using the sup-norm on functions.
What's going on with this definition is that it matches up with how, if a probability distribution over $R$ has all but $ϵ$ of its measure on the interval $[-3,5]$, then if you take any two bounded functions $f$ and $f^{\prime }$ and have them agree on said interval, the difference in their expectation values can be at most $ϵ$ times the difference in the functions outside of said interval.

Anyways, armed with this notion of almost-support, the natural analogue of "you can find compact sets that account for as much of the measure as you want" for probability distributions becomes:

Definition 3: Compact Almost-Support
A functional $h:C_B\left(X\right)\to R$ is said to have compact almost-support if, for all $ϵ>0$, there is a compact set $C_{ϵ}\subseteq X$ that is an $ϵ$-almost support for $h$.

This condition essentially rules out having too much Knightian uncertainty, lets you apply compactness arguments, and has a very nice interpretation on the set side of LF-duality.

So, we will mandate another condition to call something an infradistribution: Compact almost-support, or CAS for short. And now, we will define infradistributions afresh:

Definition 4: Infradistribution (set form)
An infradistribution over $X$ is a subset $H$ of the cone of a-measures over $X$, fulfilling the following properties:

1: Convexity: $H=c.h\left(H\right)$

2: Closure: $H=\overline{H}$

3: Upper-completeness: $H=H+\left\{\right(0,b)|b\ge 0\}$

4: Projected-compactness: $\{m|\exists b:(m,b)\in H\}$ is relatively compact in $\Delta X$ (contained in a compact set)

5: Normalization: There exists an a-measure $({\lambda }_0{\mu }_0,b_0)$ in $H$ where $b_0=0$, and there exists an a-measure $({\lambda }_1{\mu }_1,b_1)$ in $H$ where ${\lambda }_1+b_1=1$, and all a-measures $(\lambda \mu ,b)\in H$ have $\lambda +b\ge 1$.

The main thing to note in this definition that we've swapped out our notion of upper-completion. Instead of our notion of upper-completion being "add the cone of all sa-measures", our notion of upper-completion is now just "add the cone of increasing the b term". And the notion of minimal point is altered accordingly, so now it's really easy to find a minimal point below something in an infradistribution set. Just decrease the b term until you can't decrease it anymore. Roughly, the reason for this is that the sa-measures were exactly the signed measures with the special property "no matter which function $f\in C(X,[0,1\left]\right)$ we use, $m\left(f\right)+b\ge 0$".However, the only $(m,b)$ points that fulfill the property "no matter which function $f\in C_B\left(X\right)$ we use, $m\left(f\right)+b\ge 0$" are those where the measure component is just 0, and $b\ge 0$, so that's the cone we use for upper completion instead.

There's also this projected-compactness property which means that if you ignore the b terms, the set of measure components is contained in a compact set. This is an important assumption to make for being able to apply our usual compactness arguments, but it's not obviously equivalent to anything we've already discussed so far. Nonobviously, however, it's actually equivalent to the conjunction of our CAS (compact almost-support) property, and Lipschitzness. We'll be using $K\left(X\right)$ for the space of compact subsets of $X$.

And, on the concave functional side of things, our conditions are:

Definition 5: Infradistribution (functional form)
An infradistribution $h$ is a functional of type $C_B\left(X\right)\to R$ fulfilling the following properties:

1: Monotonicity: $f^{\prime }\ge f\to h\left(f^{\prime }\right)\ge h\left(f\right)$

2: Concavity: $h(pf+(1-p\left)f^{\prime }\right)\ge ph\left(f\right)+(1-p)h\left(f^{\prime }\right)$

3: Normalization: $h\left(0\right)=0,h\left(1\right)=1$

4: Lipschitzness: ${sup}_{f,f^{\prime }}\frac{|h\left(f\right)-h\left(f^{\prime }\right)|}{d(f,f^{\prime })}<\infty$

5: Compact almost-support:
$\forall ϵ>0\exists C_{ϵ}\in K\left(X\right)\forall f,f^{\prime }\in C_B\left(X\right):f_{\downarrow C_{ϵ}}=f_{\downarrow C_{ϵ}}^{\prime }\to |h\left(f\right)-h\left(f^{\prime }\right)|\le ϵ\cdot d(f,f^{\prime })$

For monotonicity, the ordering on functions is $f^{\prime }\ge f$ iff $\forall x:f^{\prime }\left(x\right)\ge f\left(x\right)$. For concavity, $p\in [0,1]$. For normalization, we abuse notation a bit and use constants (like 0) as abbreviations for constant functions (like the function that maps all of $X$ to the value 0). For Lipschitzness, our notion of distance that we have is $d(f,f^{\prime })={sup}_x|f\left(x\right)-f^{\prime }\left(x\right)|$. And for compact almost-support/CAS, $C_{ϵ}$ must be a compact set.

These two different definitions are secretly isomorphic to each other. $H$ is used for the set form of an infradistribution, and $h$ is used for the expectation functional form.

Theorem 1: The set of infradistributions (set form) is isomorphic to the set of infradistributions (functional form). The $H\to h$ part of the isomorphism is given by $h\left(f\right)={inf}_{(m,b)\in H}m\left(f\right)+b$, and the $h\to H$ part of the isomorphism is given by $H=\left\{\right(m,b)|b\ge (h^{\prime }{)}^{\ast }\left(m\right)\}$, where $h^{\prime }\left(f\right)=-h(-f)$ and $(h^{\prime }{)}^{\ast }$ is the convex conjugate of $h^{\prime }$.

This is just our old LF-duality result from "Basic Inframeasure Theory", but we have to reprove it anew because we switched around the type signature on the functional side, notion of upper completion on the set side, added the compact-projection requirement on the sets, and added the Lipschitz/CAS conditions on the functionals.

Now, we can now return and define the support of an infradistribution uniquely, as just the intersection of all supports.

Proposition 2: For any infradistribution $h$ there is a unique closed set $S$ which is the intersection of all supports for $h$, and is a support itself.

This looks easy to prove. It's really not. Although it technically hasn't been proven yet, we're fairly sure that the support of an infradistribution is the closure of the union of the supports of all the measure components present in the infradistribution set. And similarly, an $ϵ$-almost-support should be a set where all measure components in the infradistribution agree that there's only $ϵ$ measure assigned outside the set. So, the notion of a support and an almost-support on the functional side of LF-duality matches up with the standard notion of a support for measures.

To recap this section, we generalize in two directions. Our first direction of generalization is going from the type signature $C(X,[0,1\left]\right)\to [0,1]$ to $C_B\left(X\right)\to R$, because it makes infradistributions behave so much more nicely, and our second direction of generalization is letting our space $X$ be Polish instead of just a compact metric space.

Polish spaces, are spaces where you can stick a metric on them, have the space be closed under limits in that metric, and have a countable dense subset, this accounts for $N$ and $R$ and other more exotic spaces. This also allows us to properly handle weird edge discontinuities in updating. To compensate for the loss of compactness, we must add an extra condition that's like "all but $ϵ$ of the behavior of the expectation functional is explained by what functions do on a compact set, for any $ϵ$".

Section 2: Types of Infradistributions

Here's our fancy diagram of implications between the various additional properties an infradistribution may have.

We should mention that a-measures can be written in the form $(\lambda \mu ,b)$. You can decompose a measure into a scale term $\lambda$ and a probability distribution $\mu$. So, if we talk about the $b$ value or the $\lambda$ value of minimal points in a set, this is what it's referring to.

Definition 6: Homogenous
An infradistribution is homogenous iff $h\left(af\right)=ah\left(f\right)$ for $a\in R^{\ge 0}$ or, equivalently, all minimal points in $H$ have $b=0$.

I haven't had much occasion to use this so far, despite it looking like a very natural condition. One notable property is that it's one of the only nice properties preserved by updating, which is an operation that's very badly behaved when it comes to preserving nice properties. Sadly, updating only preserves homogenity when $g=0$ (your off-event utility), and as we'll see later when we get to cohomogenous infradistributions, that sort of update is actually not a very good way to represent "updating when you don't care about what happens off-history". The second notable property is that, since it's equivalent to all minimal points having $b=0$, homogenous infradistributions could be thought of as corresponding to sets of actual measures. So, it's what you'd naturally get if you were fine with dealing with sets of measures instead of probability distributions, but had philosophical issues with +b offset/guaranteed utility.

Proposition 3: $h\left(af\right)=ah\left(f\right)$ for all $a\ge 0$ iff all minimal points in $H$ have $b=0$.

In the other direction, we have 1-Lipschitzness. This is as simple as it says, it's just that the infradistribution $h$ has a Lipschitz constant of 1. Or, if you prefer to think in terms of minimal points, all minimal points have $\lambda \le 1$. Their measure component is an actual probability distribution or has even less measure present than that.

The primary nice application this property has is that, intuitively, when you compose a bunch of operations, the Lipschitz constants tend to blow up over time, like how adding or multiplying Lipschitz functions often leads to the Lipschitz constant of the result increasing. So, if you want to infinitely iterate a process, this condition shows up as a prerequisite for doing that.

However, for technical reasons, 1-Lipschitzness doesn't behave as nicely for infradistributions of type $C_B\left(X\right)\to R$ as it does for infradistributions of type $C(X,[0,1\left]\right)\to [0,1]$, so in practice, we use the stronger condition of C-additivity instead.

Also, for minimal points, 1-Lipschitzness corresponds to "all minimal points have their measure component being a probability distribution or having lower measure than that". So it's what you'd naturally get if you were fine with +b offsets, but had philosophical issues with the amount of measure being more than 1. After all, you can interpret measures of less than 1 as "there's some probability of not getting to select from this probability distribution in the first place", but how do you interpret measures of more than 1?

Proposition 4: $\frac{|h\left(f\right)-h\left(f^{\prime }\right)|}{d(f,f^{\prime })}\le 1$ iff all minimal points in $H$ have $\lambda \le 1$.

There are two strengthenings of that property. Our first one will be cohomogenity, which, oddly enough, is a little nicer than homogenity to have as a property, despite being more complicated to state.

Definition 7: Cohomogenous
An infradistribution is cohomogenous iff $h(1+af)=1-a+ah(1+f)$ for all $a\ge 0$ or, equivalently, if all minimal points have $\lambda +b=1$.

Why is this called cohomogenous? Well, for homogenity, if you drew a line from 0 to $f$, and plotted what $h$ does over that line, it'd be linear. For cohomogenity, if you drew a line from 1 to $1+f$, and plotted what h does over that line, it'd be linear. The half of normalization where $h\left(0\right)=0$ enforces that there's a minimal point with $b=0$, and we can compare with homogenity being "all minimal points have $b=0$". The half of normalization where $h\left(1\right)=1$ enforces that there's a minimal point with $\lambda +b=1$, and we can compare that with cohomogenity being "all minimal points have $\lambda +b=1$". Homogenity is "the whole function $h$ is determined by the differentials at 0" and cohomogenity is "the whole function $h$ is determined by the differentials at 1".

Also, "all minimal points have $\lambda +b=1$" is what you'd naturally get if you adopted the 1-Lipschitz viewpoint of not being ok with the measure components of a-measures summing to more than 1, and adopted a view of the b component that's something like "you have probability 1 of either proceeding with the probability distribution of interest, or the experiment not starting in the first place and you go to Nirvana (1 reward Nirvana, not infinite reward)".

This property is preserved by updates but only for $g=1$, like how homogenity is preserved by updates where $g=0$. And in fact, it makes somewhat more sense to use this for updating when you don't care about what happens off-history than the $g=0$ update. Here's why. If you observe an observation, an update where $g=0$ is like "ok, we do as badly as possible if the observation doesn't happen". Thus, your expectations will tend to be determined by the probability distributions with the lowest probability of observing the event, because you're trying to salvage the worst-case.

However, an update when $g=1$ is like "we do well if the observation doesn't happen". Thus, your expectations will tend to be determined by the probability distributions with the highest probability of observing the event, which is clearly more sensible behavior. Cohomogenity is preserved under these sorts of updates.

There's also an important thing we become able to do at this stage. As we'll see at the end of this post, we can define entropy for infradistributions in general. However, it only depends on the differentials of an infradistribution around 1/the minimal points with $\lambda +b=1$. And cohomogenity is "those differentials/minimal points pin down the whole infradistribution". So, my inclination is to view the concept of entropy for infradistributions in general as a case of extending a definition further than it's meant to go, and I suspect entropy will only end up being relevant for cohomogenous infradistributions.

Proposition 5: $h(1+af)=1-a+ah(1+f)$ iff all minimal points in $H$ have $\lambda +b=1$.

Time for the next one, C-additivity, which is an incredibly useful one.

Definition 8: C-additive
An infradistribution is C-additive iff $h\left(c\right)=c$ for all $c\in R$, or, equivalently, if all minimal points have $\lambda =1$, or, equivalently, if $h(c+f)=c+h\left(f\right)$ for all $c\in R$

This lets you pull constants out of functions, a very handy thing to do. For sets, it says that every minimal point can be represented as a probability distribution paired with a b term. You'd get this if, philosophically, you were fine with +b guaranteed utility, but were a real stickler for everything being a probability distribution. Further, you need to assume this property to get obvious-looking properties of products to work, like "projecting a semidirect product on $X\times Y$ back to $X$ makes your original infradistribution" or "projecting a direct product to either coordinate makes the relevant infradistribution" or "projecting the free product to either coordinate makes the relevant infradistribution". So, this could be thought of as the property that makes products work sensibly, which explains why it shows up so often. In particular, though we don't know whether it's necessary to make things like the infinite semidirect product work out, it's certainly sufficient to do so.

Proposition 6: $h(c+f)=c+h\left(f\right)$ iff all minimal points in $H$ have $\lambda =1$ iff $h\left(c\right)=c$.

And now we get to the single most important and nicely behaved property of all of these, crispness.

Definition 9: Crisp
An infradistribution is crisp iff $h(c+af)=c+ah\left(f\right)$ for $c\in R$, $a\ge 0$, or, equivalently, if all minimal points have $\lambda =1$ and $b=0$.

So, this is really dang important, mostly because of that property where all minimal points have $\lambda =1$ and $b=0$. It means all crisp infradistributions can be viewed as a compact convex set of probability distributions! These are already studied quite extensively in the area of imprecise probability. Shige Peng's nonlinear expectation functionals correspond to these, as do the "lower previsions" from the textbook "Statistical Reasoning with Imprecise Probabilities".

Any two out of three from "homogenity", "cohomogenity" and "C-additivity" imply this property, and it implies all three. Homogenity and 1-Lipschitzness also suffice to guarantee this condition.

Now, the ability to identify these with conventional sets of probability distributions lets you simplify things considerably. They have a natural interpretation of entropy, they make very nice choices for infrakernels in MDP's and POMDP's, and (we won't cover this in this post), they let you break down time-discounted utility functions in a very handy way, because the reward that has occured so far is a constant and you can pull that out, and you can factor a time-discount out of everything else to pull the time-discount scaling factor out of the expectation as well. So these are very nice.

Proposition 7: $h(c+af)=c+ah\left(f\right)$ iff all minimal points in $H$ have $\lambda =1$ and $b=0$.

Time for one last property, which is the analogue for infradistributions of dirac-delta distributions in standard probability theory.

Definition 10: Sharp
An infradistribution is sharp iff $h\left(f\right)={inf}_{x\in C}f\left(x\right)$ for some compact set $C\subseteq X$, or, equivalently, if the set of minimal points in $H$ is all probability distributions supported on $C$.

Notice that $C$ doesn't have to be a convex set, just compact. $C$ is a subset of $X$, not a subset of the space of a-measures. These are infradistributions which correspond to "Murphy can pick the worst possible point from this set right here and that's all they can do". They're like the infradistribution analogue of probability distributions that put all their probability on a single point. This analogy is furthered by probability distributions supported on a single point (the dirac-delta distributions) being extreme points in the space of probability distributions, and we have a similar result here. An extreme point is a point in a set that cannot be made by taking probabilistic mixtures of any distinct points in the set.

Proposition 8: $h\left(f\right)={inf}_{x\in C}f\left(x\right)$ iff the set of minimal points of $H$ corresponds to the set of probability distributions supported on $C$.

Proposition 9: All sharp infradistributions are extreme points in the space of crisp infradistributions.

These are almost certainly not the only extreme points in the space of crisp infradistributions, but they are extreme points nonetheless.

And that should be about all of it. Now for further things we can do with infradistributions!

Section 3: Operations on Infradistributions

We start by giving a big chart for which operations preserve which properties.

This diagram doesn't show everything in this section, it's missing infrakernel pushforward, Markov kernel pushforward, direct product, coproduct, free product, and probably more. However, all of those can be generated via composing continuous pushforward, inf, sup, continuous pullback, and semidirect product accordingly, so they automatically preserve the nice properties because their building blocks do. Since we won't be addressing continuous pushforward, mixture, and updating in this section (we already did it in the last post), we'll just get them out of the way right now.

Proposition 10: Mixture, updating, and continuous pushforward preserve the properties indicated by the diagram, and always produce an infradistribution.

This seems like routine verification, but why do we need to specify that they always produce an infradistribution? Didn't we know that already from the last post? Well, because we're dealing with Polish spaces now, and also altered the type signature of our infradistributions, we do need to double check that our various conditions still work out. Fortunately, they do.

Our first two that we'll be looking at are inf and sup.

IMPORTANT EDIT RE: ORDERING

We've got a call to make on ordering infradistributions and would appreciate feedback in the comments.

One possible way to do it is to have the ordering on infradistributions be the same as the ordering on functionals, like $h\ge h^{\prime }$ iff $\forall f\in C_B\left(X\right):h\left(f\right)\ge h^{\prime }\left(f\right)$. This is the information ordering from Domain theory, where the least informative (most uncertainty about what Murphy does/biggest sets) go at the bottom, and the most informative (smallest sets/least uncertainty about what Murphy does) go near the top, with a missing top element representing inconsistency. Further support for this view comes from the supremum in the information ordering on infradistributions perfectly lining up with how the supremum from domain theory behaves. The problem is, the "or" operation in infradistribution logic is infinimum in the information ordering, and the way that lattice theory writes inf is $\wedge$. The bullet bitten in this approach is using the symbol $\vee$ to mean infinimum, in contravention of the rest of lattice theory, confusing many people. But, I mean, $\vee$ points down, it looks like it means inf if you don't have lattice theory familiarity. 

Or, there's Vanessa's preferred view where we use the set ordering, where the smallest sets go on the bottom, and the biggest sets go on the top. This is in perfect compliance with lattice theory, there's no notation mismatches. 

"It's extremely standard in lattice theory to denote join by $\vee$ and meet by $\wedge$. I can't imagine submitting a paper which would have it the reverse."

But it means that the ordering is backwards from domain theory (inf in the set ordering behaves like sup does in domain theory), and supremum in this ordering is the inf of two functions.

Oh, there's a third way: Use the domain theory information ordering, but use $\bigsqcup$ for supremum aka intersection aka logical and, and $\sqcap$ for infinimum aka union aka logical or. This notation is standard in domain theory, because there are often orderings or logical operations on the domains that don't match up with the order a domain is equipped with, and the lattice theorists would understand it.  This would look a little strange because we'd have $\bigsqcup =\cap =\wedge$ and $\sqcap =\cup =\vee$, but that's the only drawback.

The infinimum (information ordering) is easy to define.

Definition 11: Infinimum
The infinimum of two infradistributions is defined as:
$inf(h_1,h_2)\left(f\right):=inf\left(h_1\right(f),h_2(f\left)\right)$

Proposition 11: The inf of two infradistributions is always an infradistribution, and inf preserves the infradistribution properties indicated by the diagram at the start of this section.

Now, when we say what a construction is on the set side of LF-duality, we should remember that taking the convex hull, closure, and upper completion always gets a set into its canonical infradistribution form, without changing any of the expectations at all. It gets pretty hard to show whether the set form for the more complicated constructions preserves convex hull, closure, and upper completion, and it affects nothing whether or not it does. So, from now on, when we give a set form for something you can do with infradistributions, it should be understood to be modulo convex hull, closure, and upper completion.

Anyways, our definition of inf on the set level is

$inf(H_1,H_2):=H_1\cup H_2$

And we can show that this is the right definition (modulo closed convex hull and upper completion)

Proposition 12: $E_{inf(H_1,H_2)}\left(f\right)=inf\left(E_{H_1}\right(f),E_{H_2}(f\left)\right)$

Excellent, it matches up with the concave functional definition.

One last result. Can we take the infinimum of infinitely many infradistributions and have it be an infradistribution? Well, yes, as long as you have certain conditions in place.

Proposition 13: If a family of infradistributions $\{h_i{\}}_{i\in I}$ has a shared upper bound on the Lipschitz constant, and for all $ϵ$, there is a compact set $C_{ϵ}$ that is an $ϵ$-almost support for all $h_i$, then ${inf}_i{h}_i$, defined as $\left({inf}_i{h}_i\right)\left(f\right):={inf}_i\left(h_i\right(f\left)\right)$, is an infradistribution. Further, for all conditions listed in the table, if all the $h_i$ fulfill them, then ${inf}_i{h}_i$ fulfills the same property.

The supremum is a bit more finicky, because it might not exist. We haven't shown it yet, but if infinimum is union of infradistribution sets, then supremum should probably be intersection of infradistribution sets. So, the supremum of the infradistributions corresponding to two dirac-delta distributions on different points has no intersection, and doesn't exist. However, it's reasonable to ask what suprema are like if they do exist.

The upcoming definition may not produce an infradistribution, but fortunately we'll see we only need to check normalization to ensure that it's an infradistribution.

Definition 12: Supremum
The supremum of two infradistributions $h_1,h_2$ is defined as:
$sup(h_1,h_2)\left(f\right):={sup}_{p,f_1,f_2:p{f}_1+(1-p)f_2\le f}p{h}_1\left(f_1\right)+(1-p)h_2\left(f_2\right)$
Alternately, it is the least infradistribution greater than $h_1,h_2$ in the information ordering, or the concave monotone hull of $f\mapsto sup\left(h_1\right(f),h_2(f\left)\right)$.

Proposition 14: If $sup(h_1,h_2)\left(0\right)=0$ and $sup(h_1,h_2)\left(1\right)=1$, then the supremum is an infradistribution.

Proposition 15: All three characterizations of the supremum given in Definition 12 are identical.

What's the supremum on the set level? Well, it's:

$sup(H_1,H_2):=H_1\cap H_2$

Just take intersection. This makes it clearer why the supremum may fail to exist. Maybe the two sets have empty intersection. Or, since normalization is "there exists an a-measure with $b=0$, and all a-measures have $\lambda +b\ge 1$ and there's an a-measure with $\lambda +b=1$", you need the intersection to contain a point with $b=0$ and a point with $\lambda +b=1$ as a necessary and sufficient condition for the supremum to be an infradistribution.

Proposition 16: $E_{sup(H_1,H_2)}\left(f\right)={sup}_{p,f_1,f_2:p{f}_1+(1-p)f_2\le f}p{E}_{H_1}\left(f_1\right)+(1-p)E_{H_2}\left(f_2\right)$

Ah good, we've got the right set form. Infinimum is union, supremum is intersection. Pleasing. This view of supremum as intersection is also the easiest way to show that:

Proposition 17: For any property in the table at the start of this section, $sup(h_1,h_2)$ will fulfill the property if both components fulfill the property.

And what about infinite supremum? Well... Let's take a bit of a detour to Domain Theory. Domain Theory studies partially ordered sets with particular properties in order to give a semantics to types in computer programming, where the ordering on the set can be interpreted as an "information ordering". Lower things in the ordering are less-defined pieces of data or stages of a computation. Let's look to it for inspiration, because "partially specified information" looks very similar to the sort of stuff we're doing where we've got a set of possible a-measures that can be large (much ignorance) or small (much knowledge)

The pairwise supremum rarely exists in full generality for a domain, for it usually happens that it is possible for there to exist two pieces of inconsistent information that cannot be combined. However, one of the requirements for a domain to be a domain is that directed sets have a supremum. A directed set is a set of points $D$ where, for any two points $x,y\in D$, there is some third point $z\in D$ where $z\ge x$ and $z\ge y$. Ie, "any two pieces of information can be consistently combined". Or you could think of it as analogous to compact sets. If there's no finite collection of compact sets with nonempty intersection, then the intersection of all of them has nonempty intersection. If there's no finite collection of data that is inconsistent, then it should be possible to aggregate all the data together.

There's also a particularly nicely behaved subclass of domains called bc-domains where the only possible obstruction to the pairwise supremum existing is the lack of any upper bounds to both points, and you have a guarantee that if there's some $z$ that's an upper bound to $x$ and $y$, the the supremum of $x$ and $y$ exists. In other words, if it's possible to combine the information together at all, there's some minimal way to combine the information together.

Finally, if you have "supremum exists for any finite collection of points in this set $S$" and "supremum exists for any directed set $D$", then you can show that there's a supremum for your arbitrary set $S$ by "directifying" it. You take all possible collections of finitely many points from $S$, take the supremum of each of those, add all those points to your set $S$, and now you have created a directed set, and can take the supremum of that, and it's the least upper bound on your original set $S$.

As it turns out, we have a perfectly analogous situation for all of these concepts for infradistributions. Call a set of infradistributions $\{h_i{\}}_{i\in I}$ "directifiable" if any finite collection of infradistributions from that set has the supremum exist. Then, we get:

Proposition 18: If a family of infradistributions $h_i$ is directifiable, then ${sup}_i{h}_i$ (defined as the functional corresponding to the set ${\bigcap }_i{H}_i$) exists and is an infradistribution. Further, for all conditions listed in the table, if all the $h_i$ fulfill them, then ${sup}_i{h}_i$ fulfills the same property.

One last note of relevance. For infradistributions of type $C(X,[0,1\left]\right)\to [0,1]$, it is possible to associate them with a canonical infradistribution of type $C_B\left(X\right)\to R$ by taking the maximal extension of the function $h$ to all of $C_B\left(X\right)$ that still fulfills all the infradistribution properties.

However, sadly, this way of embedding infradistributions of type $C(X,[0,1\left]\right)\to [0,1]$ into the poset of infradistributions of type $C_B\left(X\right)\to R$ doesn't make a sublattice. The inf of two maximal extensions may not be a maximal extension. Infradistributions of type $C_B\left(X\right)\to R$ are just much better-behaved when it comes to inf and sup.

Let's move forward. As a brief recap of "Basic Inframeasure Theory", pushforward w.r.t a a continuous function $g:X\to Y$ is the function $g_{\ast }:\square X\to \square Y$ given by: $\left(g_{\ast }\right(h\left)\right)\left(f\right)=h(f\circ g)$. Ie, if you have a continuous function $g:X\to Y$, and Murphy is adversarially choosing things to minimize functions $X\to R$, then given a function $f:Y\to R$, you can just precompose it with $g$ to turn it into a function $X\to R$, and look at the worst-case expected value of it.

Interestingly enough, a bunch of other concepts we're going to be covering can be thought of as generated by semidirect product and pushforward w.r.t a continuous function.

Next up on the list is semidirect product, which is a very important construction, and this requires rigorously defining infrakernels. Just as a Markov kernel is a function $X\to \Delta Y$, a probabilistic function from $X$ to $Y$, an infrakernel is a function $X\to \square Y$ fulfilling some special properties. Ie, Murphy has their choice of how to minimize things depend on the initial point selected. We're in Polish spaces, so we'll need to add one extra condition to ensure that infrakernels behave nicely.

Definition 13: Infrakernels
An infrakernel is a function $X\to \square Y$ that fulfills the following three properties. From now on, $X\stackrel{ik}{\to }Y$ will denote the type of infrakernels from $X$ to $\square Y$. $C_X$ and $C_{ϵ}$ denote compact subsets of $X$ and $Y$ respectively.

Condition 1: Boundedness.
${sup}_{f,f^{\prime },x}\frac{|K\left(x\right)\left(f\right)-K\left(x\right)\left(f^{\prime }\right)|}{d(f,f^{\prime })}<\infty$

Condition 2: Pointwise convergence.
$\forall f,\{x_n{\}}_{n\in N},x:{lim}_{n\to \infty }{x}_n=x\to {lim}_{n\to \infty }K(x_n\left)\right(f)=K(x\left)\right(f)$

Condition 3: Compact-shared CAS.
$\forall C_X\in K\left(X\right),ϵ\exists C_{ϵ}\in K\left(Y\right)\forall f,f^{\prime },x\in C_X:f_{\downarrow C_{ϵ}}=f_{\downarrow C_{ϵ}}^{\prime }\to |K\left(x\right)\left(f\right)-K\left(x\right)\left(f^{\prime }\right)|\le ϵd(f,f^{\prime })$

The first condition says that there's an upper bound on the Lipschitz constant of all the $K\left(x\right)$.

The second condition says that if the input to the kernel converges, that had better lead to convergence in what the corresponding infradistributions think about the expectation of a particular function.

The third condition says that, given any compact subset $C_X$ of $X$, there needs to be a sequence of compact subsets of $Y$ that act as compact almost-supports for any of the infradistributions from the family produced by feeding a point $x$ from $C_X$ into the kernel.

Roughly, these three conditions are how the infrakernel is glued together to behave nicely without the Lipschitz constant getting unmanageable, or having any discontinuities in functions, and while preserving the compactness properties we need. 

Amazingly enough, once we define a notion of distance for infradistributions later, we'll see that we can almost replace this by one single condition: "the function $K:X\to \square Y$ is bounded and continuous". So you could just consider an infrakernel to be the analogue of a Feller-continuous Markov kernel. These conditions are almost all of the way towards being continuity in disguise, despite the concept of infrakernels coming well before we started to think about distance metrics on infradistributions. But that's for later.

Now that we see what the conditions for an infrakernel mean, we can look at the semidirect product.

Definition 14: Semidirect Product
The semidirect product $⋉$, of type $\square X\times \left(X\stackrel{ik}{\to }Y\right)\to \square (X\times Y)$, is defined as
$(h⋉K)\left(f\right):=h(\lambda x.K(x\left)\right(\lambda y.f(x,y)\left)\right)$

Intuitively, the semidirect product definition is, if $x$ is locked in and known, then Murphy gets to use $K\left(x\right)$ to adversarially pick the $y$ to minimize $f(x,y)$. But, backing up, Murphy actually has adversarial choice of the $x$. It knows that picking $x$ leads to an expected payoff of $K\left(x\right)(\lambda y.f(x,y\left)\right)$, so, Murphy actually minimizes the function $\lambda x.K\left(x\right)(\lambda y.f(x,y\left)\right)$ to set itself up for success when the $x$ is revealed.

Semidirect products are important for two reasons. The first reason is that they let us define the direct product and infrakernel pushforwards. In fact, even mixture of infradistributions to make a prior can be viewed as special case of this. We'll get more into how that works later in the "infrakernel pushforward" section, but if you want an "infraprior" where Murphy has some adversarial choice over which hypothesis is selected, semidirect product is the key tool to let you figure out exactly how it should behave! Thinking about semidirect products lets you figure out the update rule when you have Knightian uncertainty about the prior probability distribution! Anything involving stacking adversarial choice over multiple stages involves semidirect product, it's an absolutely indispensable tool. Like going from Markov Decision Processes to infradistributions over histories... Yeah, that involves repeated semidirect product to build up the history.

The second reason the semidirect product is important that it gives you conditional probability. Given an input, the kernel gives you an infradistribution over what happens next, a sort of "conditional infradistribution" given a particular input. This reaches its full potential in the Infra-Disintegration Theorem, coming up in a while.

Time for results.

Proposition 19: $h⋉K$ is an infradistribution, and preserves all properties indicated in the diagram at the start of this section if h and all the $K\left(x\right)$ have said property.

This is nice to know. There's also a really basic property you'd want, though. Projecting the semidirect product back to the starting $X$ coordinate should recover your original infradistribution. However, we need C-additivity for all the $K\left(x\right)$ in order for this to work. Ie, for all $x$, $K\left(x\right)\left(c\right)=c$.