Previous proof post is here.

Theorem 2: Isomorphism Theorem: For (causal, pseudocausal, acausal, surcausal) ${\Theta }^{st}$ or ${\Theta }^{\omega }$ which fulfill finitary or infinitary analogues of all the defining conditions, $\uparrow \left({\Theta }^{st}\right)$ and $\downarrow \left({\Theta }^{\omega }\right)$ are (causal, pseudocausal, acausal, surcausal) hypotheses. Also, $\uparrow$ and ${\to }^{st}$ define an isomorphism between $\Theta$ and ${\Theta }^{st}$, and $\downarrow$ and ${\to }^{\omega }$ define an isomorphism between $\Theta$ and ${\Theta }^{\omega }$.

Proof sketch: The reason this proof is so horrendously long is that we've got almost a dozen conditions to verify, and some of them are quite nontrivial to show and will require sub-proof-sketches of their own! Our first order of business is verifying all the conditions for a full belief function for $\uparrow \left({\Theta }^{st}\right)$. Then, we have to do it all over again for $\downarrow \left({\Theta }^{\omega }\right)$. That comprises the bulk of the proof. Then, we have to show that taking a full belief function $\Theta$ and restricting it to the infinite/finite levels fulfills the infinite/finite analogues of all the defining conditions for a belief function on policies or policy-stubs, which isn't quite as bad. Once we're done with all the legwork showing we can derive all the conditions from each other, showing the actual isomorphism is pretty immediate from the Consistency condition of a belief function.

Part 1:Let's consider $\uparrow \left({\Theta }^{{\pi }_{st}}\right)$. This is defined as: $\uparrow \left({\Theta }^{st}\right)\left({\pi }_{pa}\right):={\bigcap }_{{\pi }_{st}\le {\pi }_{pa}}\left(p{r}_{\ast }^{{\pi }_{pa},{\pi }_{st}}{)}^{-1}\right({\Theta }^{st}\left({\pi }_{st}\right))$

We'll show that all 9+2 defining conditions for a belief function are fulfilled for $\uparrow \left({\Theta }^{st}\right)$. The analogue of the 9+2 conditions for a ${\Theta }_{st}$ is:

1: Stub Nirvana-free Nonemptiness: $\forall {\pi }_{st}:{\Theta }^{st}\left({\pi }_{st}\right)\cap NF\ne \varnothing$

2: Stub Closure: $\forall {\pi }_{st}:{\Theta }^{st}\left({\pi }_{st}\right)=\overline{{\Theta }^{st}\left({\pi }_{st}\right)}$

3: Stub Convexity. $\forall {\pi }_{st}:{\Theta }^{st}\left({\pi }_{st}\right)=c.h\left({\Theta }^{st}\right({\pi }_{st}\left)\right)$

4: Stub Nirvana-free Upper-Completion.

$\forall {\pi }_{st}:{\Theta }^{st}\left({\pi }_{st}\right)\cap NF=\left(\right({\Theta }^{st}\left({\pi }_{st}\right)\cap NF)+M^{sa}(F^{NF}\left({\pi }_{st}\right)\left)\right)\cap M^a\left(F\right({\pi }_{st}\left)\right)$

5: Stub Restricted Minimals:

$\exists {\lambda }^{\odot },b^{\odot }\forall {\pi }_{st}:(\lambda \mu ,b)\in {\left({\Theta }^{st}\right({\pi }_{st}\left)\right)}^{min}\to \lambda +b\le {\lambda }^{\odot }+b^{\odot }$

6: Stub Normalization: ${inf}_{{\pi }_{st}}{E}_{{\Theta }^{st}\left({\pi }_{st}\right)}\left(0\right)=0$ and ${sup}_{{\pi }_{st}}{E}_{{\Theta }^{st}\left({\pi }_{st}\right)}\left(1\right)=1$

7: Weak Consistency: $\forall {\pi }_{st}^{lo},{\pi }_{st}^{hi}\ge {\pi }_{st}^{lo}:p{r}_{\ast }^{{\pi }_{st}^{hi},{\pi }_{st}^{lo}}\left({\Theta }^{st}\right({\pi }_{st}^{hi}\left)\right)\subseteq {\Theta }^{st}\left({\pi }_{st}^{lo}\right)$

8: Stub Extreme Point Condition: for all $M,{\pi }_{st}^{lo}$:

$M\in \left({\Theta }^{st}\right({\pi }_{st}^{lo}){)}^{xmin}\cap NF\to$

$\exists \pi \ge {\pi }_{st}^{lo}\forall {\pi }_{st}^{hi}:(\pi >{\pi }_{st}^{hi}\ge {\pi }_{st}^{lo}\to (\exists M^{\prime }\in {\Theta }^{st}\left({\pi }_{st}^{hi}\right)\cap NF:p{r}_{\ast }^{{\pi }_{st}^{hi},{\pi }_{st}^{lo}}\left(M^{\prime }\right)=M\left)\right)$

9: Stub Uniform Continuity: The function ${\pi }_{st}\mapsto \left(p{r}_{\ast }^{\infty ,{\pi }_{st}}{)}^{-1}\right({\Theta }^{st}\left({\pi }_{st}\right)\cap NF\cap \{\le \odot \})$ is uniformly continuous.

C: Stub Causality: $\forall {\pi }_{st},M\in {\Theta }^{st}\left({\pi }_{st}\right)\exists of\forall {\pi }_{st}^{\prime }:of\left({\pi }_{st}\right)=M\wedge of\left({\pi }_{st}^{\prime }\right)\in {\Theta }^{st}\left({\pi }_{st}^{\prime }\right)$

where the outcome function $of$ is defined over all stubs.

P: Stub Pseudocausality: 

$\forall {\pi }_{st},{\pi }_{st}^{\prime }:\left(\right(M\in {\Theta }^{st}\left({\pi }_{st}\right)\wedge \text{supp}\left(M\right)\subseteq F^{NF}\left({\pi }_{st}^{\prime }\right))\to M\in {\Theta }^{st}({\pi }_{st}^{\prime }\left)\right)$

Let's begin showing the conditions. But first, note that since we have weak consistency, we can invoke Lemma 6 to reexpress $\uparrow \left({\Theta }^{st}\right)\left({\pi }_{pa}\right)$ as ${\bigcap }_n\left(p{r}_{\ast }^{{\pi }_{pa},{\pi }_{pa}^n}{)}^{-1}\right({\Theta }^{st}\left({\pi }_{pa}^n\right))$ Where ${\pi }_{pa}^n$ is the n'th member of the fundamental sequence of ${\pi }_{pa}$.

Also note that, for all stubs, $\uparrow \left({\Theta }^{st}\right({\pi }_{st}\left)\right)={\Theta }^{st}\left({\pi }_{st}\right)$. We'll be casually invoking this all over the place and won't mention it further.

Proof: By Lemma 6 with weak consistency, $\uparrow \left({\Theta }^{st}\right({\pi }_{st}\left)\right)={\bigcap }_{n\ge m}\left(p{r}_{\ast }^{{\pi }_{st},{\pi }_{st}^n}{)}^{-1}\right({\Theta }^{st}\left({\pi }_{st}^n\right))$ Now, m can be anything we like, as long as it's finite. Set m to be larger than the maximum timestep that the stub is defined for. Then ${\pi }_{st}^n={\pi }_{st}$ no matter what n is (since it's above m) and projection from a stub to itself is identity, so the preimage is exactly our original set ${\Theta }^{st}\left({\pi }_{st}\right)$.

We'll also be using another quick result. For all stubs ${\pi }_{st}^{hi}\ge {\pi }_{st}^{lo}$, given stub causality,

$p{r}_{\ast }^{{\pi }_{st}^{hi},{\pi }_{st}^{lo}}\left({\Theta }^{st}\right({\pi }_{st}^{hi}\left)\right)={\Theta }^{st}\left({\pi }_{st}^{lo}\right)$

Proof: Fix an arbitrary point $M\in {\Theta }^{st}\left({\pi }_{st}^{lo}\right)$. By causality, we get an outcome function which includes $M$, giving us that there's something in ${\Theta }^{st}\left({\pi }_{st}^{hi}\right)$ that projects down onto $M$. Use weak consistency to get the other subset direction.

Condition 1: Nirvana-free Nonemptiness.

Invoking Stub Nirvana-free Nonemptiness, $\forall {\pi }_{st}:{\Theta }^{st}\left({\pi }_{st}\right)\cap NF\ne \varnothing$ so we get nirvana-free nonemptiness for $\uparrow \left({\Theta }^{st}\right)\left({\pi }_{st}\right)$.

Now, assume ${\pi }_{pa}$ is not a stub. By stub-bounded-minimals, there is some ${\lambda }^{\odot }+b^{\odot }$ bound on the set of minimal points, regardless of stub. Let $\left({\Theta }^{st}\right({\pi }_{pa}^n){)}^{clip}$ be ${\Theta }^{st}\left({\pi }_{pa}^n\right)\cap \{\le \odot \}\cap NF$

This contains all the minimal nirvana-free points for ${\pi }_{pa}^n$. This set is nonempty because we have stub nirvana-free nonemptiness, so a nirvana-free point $M$ exists. We have stub-closure and stub minimal-boundedness, so we can step down to a minimal nirvana-free point below $M$, and it obeys the ${\lambda }^{\odot }+b^{\odot }$ bound.

Further, by weak consistency and projection preserving $\lambda$ and $b$ and nirvana-freeness,

$p{r}_{\ast }^{{\pi }_{pa}^{n+1},{\pi }_{pa}^n}\left(\right({\Theta }^{st}\left({\pi }_{pa}^{n+1}\right){)}^{clip})\subseteq \left({\Theta }^{st}\right({\pi }_{pa}^n){)}^{clip}$

Invoking Lemma 9, the intersection of preimages of these is nonempty. It's also nirvana-free, because if there's nirvana somewhere, it occurs after finite time, so projecting down to some sufficiently large finite stage preserves the presence of Nirvana, but then we'd have a nirvana-containing point in a nirvana-free set $\left({\Theta }^{st}\right({\pi }_{pa}^n){)}^{clip}$, which is impossible. This is also a subset of the typical intersection of preimages used to define $\uparrow \left({\Theta }^{st}\right)\left({\pi }_{pa}\right)$. Pick an arbitrary point in said intersection of preimages of clipped subsets.

Bam, we found a nirvana-free point in $\uparrow \left({\Theta }^{st}\right)\left({\pi }_{pa}\right)$ and we're done.

Time for conditions 2 and 3, Closure and Convexity. These are easy.

$\uparrow \left({\Theta }^{st}\right)\left({\pi }_{pa}\right)={\bigcap }_n\left(p{r}_{\ast }^{{\pi }_{pa},{\pi }_{pa}^n}{)}^{-1}\right({\Theta }^{st}\left({\pi }_{pa}^n\right))$

The preimage of a closed set (stub-closure) is a closed set, and the intersection of closed sets is closed, so we have closure.

Also, $p{r}_{\ast }^{{\pi }_{pa},{\pi }_{st}^n}$ is linear, so the preimage of a convex set (stub-convexity) is convex, and we intersect a bunch of convex sets so it's convex as well.

Condition 4: Nirvana-free upper completion. 

Let $M\in \uparrow \left({\Theta }^{st}\right)\left({\pi }_{pa}\right)\cap NF$. Let's check whether $M+M^{\ast }$ (assuming that's an a-measure and $M^{\ast }$ is nirvana-free) also lies in the set. A sufficient condition on this given how we defined things is that for all ${\pi }_{pa}^n$, $p{r}_{\ast }^{{\pi }_{pa},{\pi }_{pa}^n}(M+M^{\ast })\in {\Theta }^{st}\left({\pi }_{pa}^n\right)$, as that would certify that $M+M^{\ast }$ is in all the preimages.

$p{r}_{\ast }^{{\pi }_{pa},{\pi }_{pa}^n}$ is linear, so $p{r}_{\ast }^{{\pi }_{pa},{\pi }_{pa}^n}(M+M^{\ast })=p{r}_{\ast }^{{\pi }_{pa},{\pi }_{pa}^n}\left(M\right)+p{r}_{\ast }^{{\pi }_{pa},{\pi }_{pa}^n}\left(M^{\ast }\right)$

The first component is in ${\Theta }^{st}\left({\pi }_{st}\right)$, obviously. And then, by stub nirvana-free-upper-completion, we have a nirvana-free a-measure plus a nirvana-free sa-measure (projection preserves nirvana-freeness), making a nirvana-free a-measure (projection preserves a-measures), so $p{r}_{\ast }^{{\pi }_{pa},{\pi }_{pa}^n}\left(M\right)+p{r}_{\ast }^{{\pi }_{pa},{\pi }_{pa}^n}\left(M^{\ast }\right)$ is in ${\Theta }^{st}\left({\pi }_{pa}^n\right)\cap NF$, and we're done.

Condition 5: Bounded-Minimals

So, there is a critical ${\lambda }^{\odot }+b^{\odot }$ value by restricted-minimals for ${\Theta }^{st}$

Fix a ${\pi }_{pa}$, and assume that there is a minimal point $M$ in $\uparrow \left({\Theta }^{st}\right)\left({\pi }_{pa}\right)$ with a $\lambda +b$ value that exceeds the bound. Project $M$ down into each ${\Theta }^{st}\left({\pi }_{pa}^n\right)$. Projection preserves $\lambda$ and $b$ so each of these projected points $M_n$ lie above some $M_n^{min}$.

Now, invoke Lemma 7 to construct a $M_n^{\prime }\in M^a\left(F\right({\pi }_{pa}\left)\right)$ (or the nirvana-free variant) that lies below $M$, and projects down to $M_n^{min}$. Repeat this for all n. All these $M_n^{\prime }$ points are a-measures and have the standard ${\lambda }^{\odot }+b^{\odot }$ bound so they all lie in a compact set and we can extract a convergent subsequence, that converges to $M^{\prime }$, which still obeys the ${\lambda }^{\odot }+b^{\odot }$ bound.

$M^{\prime }$ is below $M$ because $\left(\right\{M\}-M^{sa}(F\left({\pi }_{pa}\right)\left)\right)\cap M^a\left(F\right({\pi }_{pa}\left)\right)$ (or the nirvana-free variant) is a closed set. Further, by Lemma 10, $M^{\prime }$ is in the defining sequence of intersections for $\uparrow \left({\Theta }^{st}\right)\left({\pi }_{pa}\right)$. This witnesses that $M$ isn't minimal, because we found a point below it that actually obeys the bounds. Thus, we can conclude minimal-point-boundedness for $\uparrow \left({\Theta }^{st}\right)$.

Condition 6: Normalization. We'll have to go out of order here, this can't be shown at our current stage. We're going to have to address Hausdorff continuity first, then consistency, and solve normalization at the very end. Let's put that off until later, and just get extreme points.

Condition 8: Extreme point condition: 

The argument for this one isn't super-complicated, but the definitions are, so let's recap what condition we have and what condition we're trying to get.

Condition we have: for all $M,{\pi }_{st}^{lo}$:

$M\in \left({\Theta }^{st}\right({\pi }_{st}^{lo}){)}^{xmin}\cap NF\to$

$\exists \pi \ge {\pi }_{st}^{lo}\forall {\pi }_{st}^{hi}:(\pi >{\pi }_{st}^{hi}\ge {\pi }_{st}^{lo}\to (\exists M^{\prime }\in {\Theta }^{st}\left({\pi }_{st}^{hi}\right)\cap NF:p{r}_{\ast }^{{\pi }_{st}^{hi},{\pi }_{st}^{lo}}\left(M^{\prime }\right)=M\left)\right)$

Condition we want: for all $M,{\pi }_{st}$,

$M\in \left({\Theta }^{st}\right({\pi }_{st}){)}^{xmin}\cap NF\to \exists \pi >{\pi }_{st},M^{\prime }:M^{\prime }\in \uparrow ({\Theta }^{st}\left)\right(\pi )\cap NF\wedge p{r}_{\ast }^{\pi ,{\pi }_{st}}(M^{\prime })=M$

Ok, so $M\in \left({\Theta }^{st}\right({\pi }_{st}){)}^{xmin}\cap NF$ By the stub extreme point condition, there's a $\pi \ge {\pi }_{st}$, where, for all ${\pi }_{st}^{hi}$ that fulfill $\pi >{\pi }_{st}^{hi}\ge {\pi }_{st}$, there's a $M^{\prime }\in {\Theta }^{st}\left({\pi }_{st}^{hi}\right)\cap NF$, where $p{r}_{\ast }^{{\pi }_{st}^{hi},{\pi }_{st}}\left(M^{\prime }\right)=M$.

Lock in the $\pi$ we have. We must somehow go from this to a $M^{\prime }\in \Theta \left(\pi \right)\cap NF$ that projects down to our point of interest. To begin with, let ${\pi }^n$ be the n'th member of the fundamental sequence for $\pi$. Past a certain point m, these start being greater than ${\pi }_{st}$. The $M^{\prime }\in {\Theta }^{st}\left({\pi }^n\right)\cap NF$ which projects down to $M$ that we get by the stub-extreme-point condition will be called $M_n^{{}^{\prime }lo}$. Pick some random-ass point in $\left(p{r}_{\ast }^{\pi ,{\pi }_{st}^n}{)}^{-1}\right(M_n^{{}^{\prime }lo})$ and call it $M_n^{{}^{\prime }hi}$.

$M_n^{{}^{\prime }hi}$ all obey the $\lambda$ and $b$ values of $M$, because it projects down to $M$. We get a limit point of them, $M^{\prime }$, and invoking Lemma 10, it's also in $\uparrow \left({\Theta }^{st}\right)\left(\pi \right)$. It also must be nirvana-free, because it's a limit of points that are nirvana-free for increasingly late times. It also projects down to $M$ because the sequence $M_n^{{}^{\prime }hi}$ was wandering around in the preimage of $M$, which is closed.

Condition 9: Hausdorff Continuity: 

Ok, this one is going to be fairly complicated. Remember, our original form is:

"The function ${\pi }_{st}\mapsto \left(p{r}_{\ast }^{\infty ,{\pi }_{st}}{)}^{-1}\right({\Theta }^{st}\left({\pi }_{st}\right)\cap NF\cap \{\le \odot \})$ is uniformly continuous"

And the form we want is:

"The function ${\pi }_{pa}\mapsto \left(p{r}_{\ast }^{\infty ,{\pi }_{pa}}{)}^{-1}\right(\uparrow \left({\Theta }^{st}\right)\left({\pi }_{pa}\right)\cap NF\cap \{\le \odot \})$ is uniformly continuous"

Uniform continuity means that if we want an $ϵ$ Hausdorff-distance between two preimages, there's a $\delta$ distance between partial policies that suffices to produce that. To that end, fix our $ϵ$. We'll show that the $\delta$ we get from uniform continuity on stubs suffices to tell us how close two partial policies must be.

So, we have an $ϵ$. For uniform continuity, we need to find a $\delta$ where, regardless of which two partial policies ${\pi }_{pa}$ and ${\pi }_{pa}^{\prime }$ we select, as long as they're $\delta$ or less apart, the sets $\left(p{r}_{\ast }^{\infty ,{\pi }_{pa}}{)}^{-1}\right(\uparrow \left({\Theta }^{st}\right)\left({\pi }_{pa}\right)\cap NF\cap \{\le \odot \})$ (and likewise for ${\pi }_{pa}^{\prime }$) are only $ϵ$ apart. So, every point in the first preimage must have a point in the second preimage only $ϵ$ distance away, and vice-versa. However, we can swap ${\pi }_{pa}$ and ${\pi }_{pa}^{\prime }$ (our argument will be order-agnostic) to establish the other direction, so all we really need to do is to show that every point in the preimage associated with ${\pi }_{pa}$ is within $ϵ$ of a point in the preimage associated with ${\pi }_{pa}^{\prime }$.

First, $m:={\log }_{\gamma }\left(\delta \right)$ is the time at which two partial policies $\delta$ apart may start differing. Conversely, any two partial policies which only disagree at-or-after m are $\delta$ apart or less. Let ${\pi }_{st}^{\ast }$ be the policy stub defined as follows: take the inf of ${\pi }_{pa}$ and ${\pi }_{pa}^{\prime }$ (the partial policy which is everything they agree on, which is going to perfectly mimic both of them up till time m), and clip things off at time m to make a stub. This is only $\delta$ apart from ${\pi }_{pa}$ and ${\pi }_{pa}^{\prime }$, because it perfectly mimics both of them up till time m, and then becomes undefined (so there's a difference at time m) Both ${\pi }_{pa}$ and ${\pi }_{pa}^{\prime }$ are $\ge {\pi }_{st}^{\ast }$.

Let $M$ be some totally arbitrary point in $\left(p{r}_{\ast }^{\infty ,{\pi }_{pa}}{)}^{-1}\right({\Theta }^{st}\left({\pi }_{pa}\right)\cap NF\cap \{\le \odot \})$. $M$ is also in $\left(p{r}_{\ast }^{\infty ,{\pi }_{st}^{\ast }}{)}^{-1}\right({\Theta }^{st}\left({\pi }_{st}^{\ast }\right)\cap NF\cap \{\le \odot \})$, because $M$ projects down to some point in ${\Theta }^{st}\left({\pi }_{st}^{\ast }\right)$ that's nirvana-free.

Let ${\pi }_{pa}^{{}^{\prime }n}$, where $n\ge m$, be the n'th stub in the fundamental sequence for ${\pi }_{pa}^{\prime }$. These form a chain starting at ${\pi }_{st}^{\ast }$ and ascending up to ${\pi }_{pa}^{\prime }$, and are all $\delta$ distance from ${\pi }_{st}^{\ast }$.

Anyways, in $M^a\left(\infty \right)$, we can make a closed ball $B_{\le ϵ}$ of size $ϵ$ around $M$. This restricts $\lambda$ and $b$ to a small range of values, so we can use the usual arguments to conclude that $B_{\le ϵ}$ is compact.

Further, because ${\pi }_{st}^{\ast }$ is $\delta$ or less away from ${\pi }_{pa}^{{}^{\prime }n}$, the two sets $\left(p{r}_{\ast }^{\infty ,{\pi }_{st}^{\ast }}{)}^{-1}\right({\Theta }^{st}\left({\pi }_{st}^{\ast }\right)\cap NF\cap \{\le \odot \})$ and $\left(p{r}_{\ast }^{\infty ,{\pi }_{pa}^{{}^{\prime }n}}{)}^{-1}\right({\Theta }^{st}\left({\pi }_{pa}^{{}^{\prime }n}\right)\cap NF\cap \{\le \odot \})$ are within $ϵ$ of each other, so there's some point of the latter set that lies within our closed $ϵ$-ball.

Consider the set ${\bigcap }_{n\ge m}\left(\right(p{r}_{\ast }^{\infty ,{\pi }_{pa}^{{}^{\prime }n}}{)}^{-1}\left({\Theta }^{st}\right({\pi }_{pa}^{{}^{\prime }n})\cap NF\cap \{\le \odot \left\}\right)\cap B_{\le ϵ})$

the inner intersection is an intersection of closed and compact sets, so it's compact. Thus, this is an intersection of an infinite family of nonempty compact sets. To check the finite intersection property, just observe that since preimages of the sets $\uparrow \left({\Theta }^{st}\right)\left({\pi }_{pa}^{{}^{\prime }n}\right)\cap NF\cap \{\le \odot \}$ get smaller and smaller as n increases due to weak-consistency but always exist.

Pick some arbitrary point $M^{\prime }$ from the intersection. it's $\le ϵ$ away from $M$ since it's in the $ϵ$-ball. However, we still have to show that $M^{\prime }$ is in $\left(p{r}_{\ast }^{\infty ,{\pi }_{pa}^{\prime }}{)}^{-1}\right(\uparrow \left({\Theta }^{st}\right)\left({\pi }_{pa}^{\prime }\right)\cap NF\cap \{\le \odot \})$ to get Hausdorff-continuity to go through.

To begin with, since $M^{\prime }$ lies in our big intersection, we can project it down to any ${\Theta }^{st}\left({\pi }_{pa}^{{}^{\prime }n}\right)\cap NF\cap \{\le \odot \}$. Projecting it down to stage n makes $M_n^{\prime }$. Let $M_{\infty }^{\prime }$ be the point in $\uparrow \left({\Theta }^{st}\right)\left({\pi }_{pa}^{\prime }\right)\cap NF\cap \{\le \odot \}$ defined by: ${\bigcap }_{n\ge m}\left(p{r}_{\ast }^{{\pi }_{pa}^{\prime },{\pi }_{st}^n}{)}^{-1}\right(M_n^{\prime })$

Well, we still have to show that this set is nonempty, contains only one point, and that it's in $\uparrow \left({\Theta }^{st}\right)\left({\pi }_{pa}^{\prime }\right)$, and is nirvana-free, to sensibly identify it with a single point.

Nonemptiness is easy, just invoke Lemma 9. It lies in the usual intersections that define $\uparrow \left({\Theta }^{st}\right)\left({\pi }_{pa}^{\prime }\right)$, so we're good there. If it had nirvana, it'd manifest at some finite point, but all finite projections are nirvana-free, so it's nirvana-free. If it had more than one point in it, they differ at some finite stage, so we can project to a finite ${\pi }_{pa}^{{}^{\prime }n}$ to get two different points, but they both project to $M_n^{\prime }$, so this is impossible. Thus, $M_{\infty }^{\prime }$ is a legit point in the appropriate set. If the projection of $M^{\prime }$ didn't equal $M_{\infty }^{\prime }$, then we'd get two different points, which differ at some finite stage, so we could project down to separate them, but they both project to $M_n^{\prime }$ for all n so this is impossible.

So, as a recap, we started with an arbitrary point $M$ in $\left(p{r}_{\ast }^{\infty ,{\pi }_{pa}}{)}^{-1}\right(\uparrow \left({\Theta }^{st}\right)\left({\pi }_{pa}\right))\cap \{\le \odot \}$, and got another point $M^{\prime }$ that's only $ϵ$ or less away and lies in $\left(p{r}_{\ast }^{\infty ,{\pi }_{pa}^{\prime }}{)}^{-1}\right(\uparrow \left({\Theta }^{st}\right)\left({\pi }_{pa}^{\prime }\right))\cap \{\le \odot \}$ This argument also works if we flip ${\pi }_{pa}$ and ${\pi }_{pa}^{\prime }$, so the two preimages are only $ϵ$ or less apart in Hausdorff-distance.

So, given some $ϵ$, there's some $\delta$ where any two partial policies which are only $\delta$ apart have preimages only $ϵ$ apart from each other in Hausdorff-distance. And thus, we have uniform continuity for the function mapping ${\pi }_{pa}$ to the set of a-measures over infinite histories which project down to $\uparrow \left({\Theta }^{st}\right)\left({\pi }_{pa}\right)\cap NF\cap \{\le \odot \}$ Hausdorff-continuity is done.

Condition 7: Consistency.

Ok, we have two relevant things to check here. The first, very easy one, is that

$\uparrow \left({\Theta }^{st}\right)\left({\pi }_{pa}\right)={\bigcap }_{{\pi }_{st}\le {\pi }_{pa}}\left(p{r}_{\ast }^{{\pi }_{pa},{\pi }_{st}}{)}^{-1}\right(\uparrow \left({\Theta }^{st}\right)\left({\pi }_{st}\right))$

From earlier, we know that $\uparrow \left({\Theta }^{st}\right)\left({\pi }_{st}\right)={\Theta }^{st}\left({\pi }_{st}\right)$, and from how $\uparrow \left({\Theta }^{st}\right)\left({\pi }_{pa}\right)$ is defined, this is a tautology.

The other, much more difficult direction, is that

$\uparrow \left({\Theta }^{st}\right)\left({\pi }_{pa}\right)=\overline{c.h}\left({\bigcup }_{\pi \ge {\pi }_{pa}}p{r}_{\ast }^{\pi ,{\pi }_{pa}}\right(\uparrow \left({\Theta }^{st}\right)\left(\pi \right)\left)\right)$

We'll split this into four stages. First, we'll show one subset direction holds in full generality. Second, we'll get the reverse subset direction for causal/surcausal. Third, we'll show it for policy stubs for pseudocausal/acausal, and finally we'll use that to show it for all partial policies for pseudocausal/acausal.

First, the easy direction. $\uparrow \left({\Theta }^{st}\right)\left({\pi }_{pa}\right)\supseteq \overline{c.h}\left({\bigcup }_{\pi \ge {\pi }_{pa}}p{r}_{\ast }^{\pi ,{\pi }_{pa}}\right(\uparrow \left({\Theta }^{st}\right)\left(\pi \right)\left)\right)$

If we pick an arbitrary $M\in \uparrow \left({\Theta }^{st}\right)\left(\pi \right)$, it projects down to ${\Theta }^{st}\left({\pi }_{st}\right)$ for all stubs ${\pi }_{st}$ below $\pi$. Since ${\pi }_{pa}\le \pi$, it projects down to all stubs beneath ${\pi }_{pa}$. Since projections commute, $M$ projected down into $M^a\left(F\right({\pi }_{pa}\left)\right)$ makes a point that lies in the preimage of all the ${\Theta }^{st}\left({\pi }_{st}\right)$ where ${\pi }_{st}\le {\pi }_{pa}$, so it projects down into $\uparrow \left({\Theta }^{st}\right)\left({\pi }_{pa}\right)$.

This holds for all points in $\uparrow \left({\Theta }^{st}\right)\left(\pi \right)$, so $p{r}_{\ast }^{\pi ,{\pi }_{pa}}(\uparrow ({\Theta }^{st}\left)\right(\pi \left)\right)\subseteq \uparrow \left({\Theta }^{st}\right)\left({\pi }_{pa}\right)$. This works for all $\pi \ge {\pi }_{pa}$, so it holds for the union, and then due to closure and convexity which we've already shown, we get that the closed convex hull of the projections lies in $\uparrow \left({\Theta }^{st}\right)\left({\pi }_{pa}\right)$ too, establishing one subset direction in full generality.

Now, for phase 2, deriving $\uparrow \left({\Theta }^{st}\right)\left({\pi }_{pa}\right)\subseteq \overline{c.h}\left({\bigcup }_{\pi \ge {\pi }_{pa}}p{r}_{\ast }^{\pi ,{\pi }_{pa}}\right(\uparrow \left({\Theta }^{st}\right)\left(\pi \right)\left)\right)$ in the causal/surcausal case.

First, observe that if $\pi \ge {\pi }_{pa}$, then ${\pi }^n\ge {\pi }_{pa}^n$. Fix some $M\in \uparrow \left({\Theta }^{st}\right)\left({\pi }_{pa}\right)$ and arbitrary $\pi \ge {\pi }_{pa}$. We'll establish the existence of a $M^{\prime }\in \uparrow \left({\Theta }^{st}\right)\left(\pi \right)$ that projects down to $M$.

To begin with, $M$ projects down to $M_n$ in ${\Theta }^{st}\left({\pi }_{pa}^n\right)$. Lock in a value for n, and consider the sequence that starts off $M_0,M_1...M_n$, and then, by causality for stubs and ${\pi }^n\ge {\pi }_{pa}^n$, you can find something in ${\Theta }^{st}\left({\pi }^n\right)$ that projects down onto $M_n$, and something in ${\Theta }^{st}\left({\pi }^{n+1}\right)$ that projects down onto that, and complete your sequence that way, making a sequence of points that all project down onto each other that climb up to $\pi$. By Lemma 10, we get a $M_n^{\prime }\in \uparrow \left({\Theta }^{st}\right)\left(\pi \right)$. You can unlock n now. All these $M_n^{\prime }$ have the same $\lambda$ and $b$ value because projection preserves them, so we can isolate a convergent subsequence converging to some $M^{\prime }\in \uparrow \left({\Theta }^{st}\right)\left(\pi \right)$.

Assume $p{r}_{\ast }^{\pi ,{\pi }_{pa}}\left(M^{\prime }\right)\ne M$. Then we've got two different points. They differ at some finite stage, so there's some n where can project down onto $M^a\left(F\right({\pi }_{pa}^n\left)\right)$ to witness the difference, but from our construction process for $M^{\prime }$, both $M^{\prime }$ and $M$ project down to $M_n$, and we get a contradiction.

So, since $p{r}_{\ast }^{\pi ,{\pi }_{pa}}(\uparrow ({\Theta }^{st}\left)\right(\pi \left)\right)=\uparrow \left({\Theta }^{st}\right({\pi }_{pa}\left)\right)$, this establishes the other direction, showing equality, and thus consistency, for causal/surcausal hypotheses.

For part 3, we'll solve the reverse direction for pseudocausal/acausal hypotheses in the case of stubs, getting $\uparrow \left({\Theta }^{st}\right)\left({\pi }_{st}\right)\subseteq \overline{c.h}\left({\bigcup }_{\pi \ge {\pi }_{pa}}p{r}_{\ast }^{\pi ,{\pi }_{st}}\right(\uparrow \left({\Theta }^{st}\right)\left(\pi \right)\left)\right)$

Since we're working in the Nirvana-free case and are working with stubs, we can wield Lemma 3

$c.h\left(\right({\Theta }^{st}\left({\pi }_{st}\right){)}^{min})=c.h\left(\right({\Theta }^{st}\left({\pi }_{st}\right){)}^{\text{xmin}})$

So, if we could just show that the union of the projections includes all the extreme minimal points, then when we take convex hull, we'd get the convex hull of the extreme minimal points, which by Lemma 3, would also nab all the minimal points as well. By Lemmas 11 and 12, our resulting convex hull of a union of projections from above would be upper-complete. It would also get all the minimal points, so it'd nabs the entire ${\Theta }^{st}\left({\pi }_{st}\right)$ within it and this would show the other set inclusion direction for pseudocausal/acausal stubs. Also, we've shown enough to invoke Lemma 20 to conclude that said convex hull is closed. Having fleshed out that argument, all we need that all extreme minimal points are captured by the union of the projections.

By our previously proved extreme minimal point condition, for every extreme minimal point $M$ in ${\Theta }^{st}\left({\pi }_{st}\right)$, there's some $\pi >{\pi }_{st}$ and $M^{\prime }$ in $\uparrow \left({\Theta }^{st}\right)\left(\pi \right)$ that projects down to $M$, which shows that all extreme points are included, and we're good.

For part 4, we'll show that in the nirvana-free pseudocausal/acausal setting, we have

$\uparrow \left({\Theta }^{st}\right)\left({\pi }_{pa}\right)\subseteq \overline{c.h}\left({\bigcup }_{\pi \ge {\pi }_{pa}}p{r}_{\ast }^{\pi ,{\pi }_{pa}}\right(\uparrow \left({\Theta }^{st}\right)\left(\pi \right)\left)\right)$

Fix some arbitrary $M\in \uparrow \left({\Theta }^{st}\right)\left({\pi }_{pa}\right)$. Our task is to express it as a limit of some sequence of points that are mixtures of stuff projected from above.

For this, we can run through the same exact proof path that was used in the part of the Lemma 21 proof about how the nirvana-free part of $\Theta \left({\pi }_{pa}\right)$ is a subset of closed convex hull of the projections of the nirvana-free parts of $\Theta \left(\pi \right)$, $\pi \ge {\pi }_{pa}$. Check back to it. Since we're working in the Nirvana-free case, we can apply it very straightforwardly. The stuff used in that proof path is the ability to project down and land in $\uparrow \left({\Theta }^{st}\right)\left({\pi }_{pa}^n\right)$ (we have that by how we defined $\uparrow$), Hausdorff-continuity (which we have), and stubs being the convex hull, not the closed convex hull, of projections of stuff above them (which we mentioned recently in part 3 of our consistency proof).

Thus, consistency is shown.

Condition 6: Normalization. 

Ok, now that we have all this, we can tackle our one remaining condition, normalization. Then move on to the two optional conditions, causality and pseudocausality.

A key thing to remember is, in this setting, when you're doing $E_H\left(f\right)$, it's actually $E_{H\cap NF}\left(f\right)$, because if Murphy picked a thing with nirvana in it, you'd get infinite value, which is not ok, so Murphy always picks a nirvana-free point.

Let's show the first part, that ${inf}_{\pi }{E}_{\uparrow \left({\Theta }^{st}\right)\left(\pi \right)}\left(0\right)=0$. This unpacks as:${inf}_{\pi }{min}_{(\lambda \mu ,b)\in \uparrow \left({\Theta }^{st}\right)\left(\pi \right)\cap NF}b=0$ and we have that ${inf}_{{\pi }_{st}}{min}_{(\lambda \mu ,b)\in {\Theta }^{st}\left({\pi }_{st}\right)\cap NF}b=0$

Projections preserves $b$ values, so we can take some nirvana-free point with a $b$ value of nearly 0, and project it down to ${\Theta }^{st}\left({\pi }_{\varnothing }\right)$ (belief function of the empty policy-stub) where there's no nirvana possible because no events happen.

So, we've got $b$ values of nearly zero in there. Do we have a point with a $b$ value of exactly zero? Yes. it's closed, and has bounded minimals, so we can go "all positive functionals are minimized by a minimal point", to get a point with a $b$ value of exactly zero. Then, we can invoke Lemma 21 (we showed consistency, extreme points, and all else required to invoke it) to decompose our point into the projections of nirvana-free stuff from above, all of which must have a $b$ value of 0. So, there's a nirvana-free point in some policy with 0 $b$ value.

Now for the other direction. Let's show that ${sup}_{\pi }{E}_{\uparrow \left({\Theta }^{st}\right)\left(\pi \right)}\left(1\right)=1$.

This unpacks as: ${sup}_{\pi }{min}_{(\lambda \mu ,b)\in \uparrow \left({\Theta }^{st}\right)\left(\pi \right)\cap NF}(\lambda +b)=1$