LBIT Proofs 3: Propositions 19-22

Diffractor

Lemma 3: If $F$ is a continuous function of type $X \to K (Y)$ , where $K (Y)$ is the space of nonempty compact subsets of the space $Y$ , then given any compact set $C^{X} \subseteq X$ , $⋃_{x \in C^{X}} F (x)$ will be compact in $Y$ .

Fix some compact set $C^{X} \subseteq X$ , and continuous function $F : X \to K (Y)$ . We will operate by taking an arbitrary open cover of $⋃_{x \in C^{X}} F (x)$ and finding a finite subcover.

Let ${O_{i}}_{i \in I}$ be an open cover of $⋃_{x \in C^{X}} F (x)$ . The $O_{i}$ are subsets of $Y$ . The topology compatible with Hausdorff distance on $K (Y)$ (space of compact subsets of $Y$ ) is the Vietoris topology, where the basis opens are given by finite collections of open sets in $Y$ . You take the set of all compact subsets of $Y$ which are subsets of the union of your finite collection of open sets, and intersect every open set in your finite collection

Accordingly, let $J = P_{f i n} (I)$ (the set of all finite subsets of $I$ , the index set for our open cover), and fix a collection of open sets in $K (Y)$ , ${O_{j}}_{j \in J}$ . The sets $O_{j}$ are defined as:

$O_{j} := {C^{Y} \in K (Y) | C^{Y} \subseteq ⋃_{i \in j} O_{i} \land \forall i \in j : C^{Y} \cap O_{i} \neq \emptyset}$

Now, all the $F (x)$ with $x \in C^{X}$ are compact ( $F$ produces compact sets as output), and they are all subsets of $⋃_{x \in C^{X}} F (x)$ , so ${O_{i}}_{i \in I}$ is a cover of $F (x)$ , and due to its compactness we can identify a finite subcover, and prune away every open st which doesn't intersect $F (x)$ . $F (x)$ is a subset of the union of those finitely many open sets, and intersects all of them, so the point $F (x) \in K (Y)$ lies in the open set $O_{j}$ induced by that finite cover of open sets.

This argument works for arbitrary $F (x)$ with $x \in C^{X}$ , so the collection ${O_{j}}_{j \in J}$ is an open cover of $F (C^{X})$ . Also, because $F$ is continuous and $C^{X}$ is compact, $F (C^{X})$ is compact, so we can identify a finite subcover from ${O_{j}}_{j \in J}$ .

Then, consider the collection of open sets $O_{i}$ where $i \in j$ for some $O_{j}$ which is part of the finite cover of $F (C^{X})$ . This is finitely many opens, we're unioning together finitely many (finitely many $O_{j}$ selected) finite sets of open sets (each $O_{j}$ is associated with finitely many $O_{i}$ that it was built from).

Now we just have to show that this collection covers $⋃_{x \in C^{X}} F (x)$ , and we'll have made our finite subcover and shown that said set is compact. Assume our finite collection of opens doesn't cover the set. Then there's some $F (x)$ which wasn't covered completely. However, the point corresponding to $F (x)$ in $K (Y)$ lies in some $O_{j}$ , and from its definition, the corresponding $O_{i}$ manage to cover $F (x)$ , and we have a contradiction. We're done.

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 $K (x)$ have said property.

To show this, we'll verify that it's well-defined at all, normalization, monotonicity, concavity, Lipschitzness, compact almost-support, and preservation of the properties.

$(h ⋉ K) (f) := h (λ x . K (x) (λ y . f (x, y)))$

Our first order of business is verifying that

$λ x . K (x) (λ y . f (x, y))$

is even a continuous function to be able to show that $h$ can accept it as input.

For continuity, let $x_{n}$ limit to $x$ , and we'll try to show that $K (x_{n}) (λ y . f (x_{n}, y))$ limits to $K (x) (λ y . f (x, y))$ . Let $λ_{K}^{⊙}$ be the Lipschitz constant upper bound of $K$ .

Pick an $ϵ$ , we'll show that there's some $m_{0}$ where

$\forall n \forall m \geq m_{0} : | K (x_{n}) (λ y . f (x_{m}, y)) - K (x_{n}) (λ y . f (x, y)) | \leq ϵ (2 | | f | | + λ_{K}^{⊙})$

First, note that ${x_{n}}_{n \in N} \cup {x}$ is a compact set because $x_{n}$ limits to $x$ . Thus, by the compact-shared compact almost-support condition on an infrakernel, there must be some compact set $C_{ϵ} \subseteq Y$ where all the $K (x_{n})$ agree that functions $f, f^{'}$ agreeing on $C_{ϵ}$ have values only $ϵ d (f, f^{'})$ apart from each other.

Now, because f is a continuous bounded function $X \times Y \to R$ , it's uniformly continuous when restricted to

$({x_{n}}_{n \in N} \cup {x}) \times C_{ϵ}$

as this is the product of two compact sets and is compact. Due to the uniform continuity of $f$ restricted to that set, there is some number $δ$ where points only $δ$ apart in that set have their values only differing by $ϵ$ . Further, there is some number $m_{0}$ where, for all $m \geq m_{0}$ , $d (x_{m}, x) < δ$ .

Additionally, the maximum difference between $λ y . f (x, y)$ and $λ y . f (x^{'}, y)$ is $2 | | f | |$ .

Now that we know our number $m_{0}$ we can pick an arbitrary $m$ above it, and go:

$\forall m \geq m_{0} \forall y \in C_{ϵ} : d ((x_{m}, y), (x, y)) = d (x_{m}, x) \leq δ$

$\forall m \geq m_{0} \forall y \in C_{ϵ} : | f (x_{m}, y) - f (x, y) | \leq ϵ$

$\forall m \geq m_{0} : d ((λ y . f (x_{m}, y))_{↓ C_{ϵ}}, (λ y . f (x, y))_{↓ C_{ϵ}}) \leq δ$

And now, because these two functions restricted to $C_{ϵ}$ are only $ϵ$ apart, we can apply Lemma 2 to conclude that (since $C_{ϵ}$ and $λ_{K}^{⊙}$ work for all the $K (x_{n})$ )

$\forall n : | K (x_{n}) (λ y . f (x_{m}, y)) - K (x_{n}) (λ y . f (x, y)) | \leq ϵ \cdot 2 | | f | | + ϵ λ_{K}^{⊙}$

This argument works for any $m \geq m_{0}$ , so we have:

$\exists m_{0} \forall n \forall m \geq m_{0} : | K (x_{n}) (λ y . f (x_{m}, y)) - K (x_{n}) (λ y . f (x, y)) | \leq ϵ (2 | | f | | + λ_{K}^{⊙})$

Letting $n = m$ in particular,

$\exists m_{0} \forall n \geq m_{0} : | K (x_{n}) (λ y . f (x_{n}, y)) - K (x_{n}) (λ y . f (x, y)) | \leq ϵ (2 | | f | | + λ_{K}^{⊙})$

And for each \eps we can construct a m_0 in this way, concluding that

${lim}_{n \to \infty} | K (x_{n}) (λ y . f (x_{x}, y)) - K (x_{n}) (λ y . f (x, y)) | = 0$

Also, from our pointwise convergence condition on infradistributions,

${lim}_{n \to \infty} K (x_{n}) (λ y . f (x, y)) = K (x) (λ y . f (x, y))$

Therefore,

${lim}_{n \to \infty} K (x_{n}) (λ y . f (x_{n}, y)) = K (x) (λ y . f (x, y))$

and so, we now know that

$λ x . K (x) (λ y . f (x, y))$

is a continuous function $X \to R$ . For boundedness, upper and lower bounds on $λ^{y} . f (x, y)$ are $| | f | |$ (and the negative version of it). Due to the shared Lipschitz constant on the $K (x)$ , an upper and lower-bound on $λ x . K (x) (λ y . f (x, y))$ is $λ_{K}^{⊙} | | f | |$ (and the negative version.) Thus, we can safely feed said function into the infradistribution $h$ , so the semidirect product is well-defined. We must still show that it makes an infradistribution.

For normalization,

$(h ⋉ K) (1) = h (λ x . K (x) (λ y .1)) = h (λ x .1) = 1$

$(h ⋉ K) (0) = h (λ x . K (x) (λ y .0)) = h (λ x .0) = 0$

For monotonicity, if $f^{'} \geq f$ ,

$\forall x : λ y . f^{'} (x, y) \geq λ y . f (x, y)$

$\forall x : K (x) (λ y . f^{'} (x, y)) \geq K (x) (λ y . f (x, y))$

$λ x . K (x) (λ y . f^{'} (x, y)) \geq λ x . K (x) (λ y . f (x, y))$

$(h ⋉ K) (f^{'}) = h (λ x . K (x) (λ y . f^{'} (x, y))) \geq h (λ x . K (x) (λ y . f^{'} (x, y))) = (h ⋉ K) (f)$

For concavity,

$(h ⋉ K) (p f + (1 - p) f^{'}) = h (λ x . K (x) (λ y . p f (x, y) + (1 - p) f^{'} (x, y)))$

$\geq h (λ x . p K (x) (λ y . f (x, y)) + (1 - p) K (x) (λ y . f^{'} (x, y)))$

$\geq p h (λ x . K (x) (λ y . f (x, y))) + (1 - p) h (λ x . K (x) (λ y . f^{'} (x, y)))$

$= p (h ⋉ K) (f) + (1 - p) (h ⋉ K) (f^{'})$

In order, this was the definition of the semidirect product, all the $K (x)$ being concave so splitting them up produces a lower value (and then monotonicity for $h$ ), then $h$ being concave.

This leaves Lipschitzness and CAS. For Lipschitzness, given some $f$ and $f^{'}$ , and letting $λ_{h}^{⊙}$ be the Lipschitz constant of $h$ , we have:

$| (h ⋉ K) (f) - (h ⋉ K) (f^{'}) | = | h (λ x . K (x) (λ y . f (x, y))) - h (λ x . K (x) (λ y . f^{'} (x, y))) |$

$\leq λ_{h}^{⊙} d (λ x . K (x) (λ y . f (x, y)), λ x . K (x) (λ y . f^{'} (x, y)))$

$= λ_{h}^{⊙} {sup}_{x} | K (x) (λ y . f (x, y)) - K (x) (λ y . f^{'} (x, y)) |$

$\leq λ_{h}^{⊙} {sup}_{x} λ_{K}^{⊙} d (λ y . f (x, y), λ y . f^{'} (x, y)) = λ_{h}^{⊙} λ_{K}^{⊙} {sup}_{x} d (λ y . f (x, y), λ y . f^{'} (x, y))$

$\leq λ_{h}^{⊙} λ_{K}^{⊙} d (f, f^{'})$

Thus, that final thing shows that there's a finite Lipschitz constant for $h ⋉ K$ .

This leaves compact almost-support. Pick any $ϵ$ . This induces a compact set $C_{ϵ}^{X}$ which is an $ϵ$ -almost-support for $h$ , and then this compact set induces a compact set $C_{ϵ}^{Y}$ which an $ϵ$ -almost-support for all the $K (x)$ where $x \in C_{ϵ}^{X}$ . Now, we can apply Lemma 2 to go:

$| h (λ x . K (x) (λ y . f (x, y))) - h (λ x . K (x) (λ y . f^{'} (x, y))) |$

$\leq ϵ d (λ x . K (x) (λ y . f (x, y)), λ x . K (x) (λ y . f^{'} (x, y)))$

$+ λ_{h}^{⊙} d ((λ x . K (x) (λ y . f (x, y)))_{↓ C_{ϵ}^{X}}, (λ x . K (x) (λ y . f^{'} (x, y)))_{↓ C_{ϵ}^{X}})$

Pretty much, that first part is the " $C_{ϵ}^{X}$ is an $ϵ$ -almost-support for $h$ " piece, and the second piece is the "hey, these two functions may be a bit different on said compact set, we've gotta multiply that by the Lipschitz constant" piece. So, let's work on unpacking these two distances. For the first one, we can go:

$d (λ x . K (x) (λ y . f (x, y)), λ x . K (x) (λ y . f^{'} (x, y)))$

$= {sup}_{x} | K (x) (λ y . f (x, y)) - K (x) (λ y . f^{'} (x, y)) |$

$\leq {sup}_{x} λ_{K}^{⊙} d (λ y . f (x, y), λ y . f^{'} (x, y))$

$= λ_{K}^{⊙} {sup}_{x} d (λ y . f (x, y), λ y . f^{'} (x, y))$

$= λ_{K}^{⊙} {sup}_{x} {sup}_{y} | f (x, y) - f^{'} (x, y) |$

$= λ_{K}^{⊙} {sup}_{x, y} | f (x, y) - f^{'} (x, y) | = λ_{K}^{⊙} d (f, f^{'})$

Substituting this back in produces:

$\leq ϵ λ_{K}^{⊙} d (f, f^{'}) + λ_{h}^{⊙} d ((λ x . K (x) (λ y . f (x, y)))_{↓ C_{ϵ}^{X}}, (λ x . K (x) (λ y . f^{'} (x, y)))_{↓ C_{ϵ}^{X}})$

Time to go after the second distance piece. We have:

$d ((λ x . K (x) (λ y . f (x, y)))_{↓ C_{ϵ}^{X}}, (λ x . K (x) (λ y . f^{'} (x, y)))_{↓ C_{ϵ}^{X}})$

$= {sup}_{x \in C_{ϵ}^{X}} | K (x) (λ y . f (x, y)) - K (x) (λ y . f^{'} (x, y)) |$

And, because $f$ and $f^{'}$ agree on $C_{ϵ}^{X} \times C_{ϵ}^{Y}$ , we have $λ y . f (x, y)$ and $λ y . f^{'} (x, y)$ agreeing on $C_{ϵ}^{Y}$ , which is an $ϵ$ -almost-support for all the $K (x)$ where $x \in C_{ϵ}^{X}$ , so we have:

$\leq {sup}_{x \in C_{ϵ}^{X}} ϵ d (λ y . f (x, y), λ y . f^{'} (x, y))$

$= ϵ {sup}_{x \in C_{ϵ}^{X}} {sup}_{y} | f (x, y) - f^{'} (x, y) |$

$\leq ϵ {sup}_{x, y} | f (x, y) - f^{'} (x, y) | = ϵ d (f, f^{'})$

Substituting this back in produces:

$\leq ϵ λ_{K}^{⊙} d (f, f^{'}) + ϵ λ_{h}^{⊙} d (f, f^{'})$

And regrouping this and recapping means that we have:

$| (h ⋉ K) (f) - (h ⋉ K) (f^{'}) | \leq ϵ (λ_{K}^{⊙} + λ_{h}^{⊙}) d (f, f^{'})$

So we have crafted a compact $ϵ (λ_{K}^{⊙} + λ_{h}^{⊙})$ -support for $h ⋉ K$ , and we can make $ϵ$ arbitrarily small, so the semidirect product has compact almost-support, which is the last condition we needed.

Time for property verification.

Homogenity:

$(h ⋉ K) (a f) = h (λ x . K (x) (λ y . a f (x, y))) = h (λ x . a K (x) (λ y . f (x, y)))$

$= a h (λ x . K (x) (λ y . f (x, y))) = a (h ⋉ K) (f)$

1-Lipschitz: We showed in the Lipschitz section that an upper bound on the Lipschitz constant of $h ⋉ K$ is the product of the Lipschitz constants of the kernel and the original infradistribution, so $1 \cdot 1 = 1$ and 1-Lipschitzness is preserved.

Cohomogenity:

$(h ⋉ K) (1 + a f) = h (λ x . K (x) (λ y .1 + a f (x, y))) = h (λ x .1 - a + a K (x) (λ y .1 + f (x, y)))$

$= h (λ x .1 + a (- 1 + K (x) (λ y .1 + f (x, y)))) = 1 - a + a h (λ x .1 - 1 + K (x) (λ y .1 + f (x, y))))$

$= 1 - a + a h (λ x . K (x) (λ y .1 + f (x, y)))) = 1 - a + a (h ⋉ K) (1 + f)$

C-additivity:

$(h ⋉ K) (c) = h (λ x . K (x) (λ y . c)) = h (λ x . c) = c$

Crispness: Both homogenity and C-additivity are preserved, so crispness is too.

Sharpness:

$(h ⋉ K) (f) = h (λ x . K (x) (λ y . f (x, y))) = h (λ x . {inf}_{y \in C_{K (x)}} f (x, y))$

$= {inf}_{x \in C_{h}} ({inf}_{y \in C_{K (x)}} f (x, y)) = {inf}_{(x, y) \in ⋃_{x \in C_{h}} ({x} \times C_{K (x)})} f (x, y)$

Our task now is to show that $⋃_{x \in C_{h}} ({x} \times C_{K (x)})$ is compact, which will take a fair amount of topology work. Our first piece that we'll need is that if $x_{n}$ limits to $x$ , then $C_{K (x_{n})}$ limits to $C_{K (x)}$ in Hausdorff-distance.

To show this, we'll split it into two parts. First, we'll assume that there is an $ϵ$ where, infinitely often, there is a point in $C_{K (x_{n})}$ that is $ϵ$ away from $C_{K (x)}$ and disprove that. Second, we'll assume there is an $ϵ$ where, infinitely often, there is a point in $C_{K (x)}$ that is $ϵ$ away from $C_{K (x_{n})}$ , and disprove that.

For the first part, assume that there is an $ϵ$ where, infinitely often, there is a point in $C_{K (x_{n})}$ that is $ϵ$ away from $C_{K (x)}$ . Craft the continuous function

$f_{1} := λ y . sup (1 - \frac{1}{ϵ} {inf}_{y^{'} \in C_{K (x)}} d (y, y^{'}), 0)$

What this does is it's 1 on the set $C_{K (x)}$ , and 0 on anything more than $ϵ$ away from it. One of our conditions on an infrakernel was that ${lim}_{n \to \infty} K (x_{n}) (f) = K (x) (f)$ , so:

${lim}_{n \to \infty} {inf}_{y \in C_{K (x_{n})}} f_{1} (y) = {inf}_{y \in C_{K (x)}} f_{1} (y)$

The latter term is 1 because $f_{1}$ is 1 over $C_{K (x)}$ . However, because we're assuming that infinitely often, there's a point in $C_{K (x_{n})}$ that is $ϵ$ away from $C_{K (x)}$ , the sequence on the left-hand side is infinitely often 0, so it doesn't converge and we have a contradiction.

For the second part, assume there is an $ϵ$ where, infinitely often, there is a point in $C_{K (x)}$ that is $ϵ$ away from $C_{K (x_{n})}$ . By compactness of $C_{K (x)}$ , we can find finitely many points $y_{i}$ in it s.t. every point in $C_{K (x)}$ is only $\frac{ϵ}{2}$ away from one of the $y_{i}$ (cover $C_{K (x)}$ with $\frac{ϵ}{2}$ -size open balls centered on points in it and take a finite subcover). Now, for each of these, we can craft a function

$f_{i} := λ y . inf (1, \frac{2}{ϵ} d (y, y_{i}))$

So, this is 0 at the point $y_{i}$ , and 1 at any distance $\frac{ϵ}{2}$ or more away from it.

One of our conditions on an infrakernel was that ${lim}_{n \to \infty} K (x_{n}) (f) = K (x) (f)$ , and there are finitely many $f_{i}$ , so there's some time where all of them nearly converge, ie:

${lim}_{n \to \infty} {sup}_{i} | K (x_{n}) (f_{i}) - K (x) (f_{i}) | = 0$

However, infinitely often there's a point $y_{n} \in C_{K (x)}$ that is $ϵ$ away from $C_{K (x_{n})}$ . $y_{n}$ is $\frac{ϵ}{2}$ away from some $y_{i}$ , so that $y_{i}$ can't be closer than $\frac{ϵ}{2}$ to $C_{K (x_{n})}$ . (if it was closer, then we could pick some point in $C_{K (x_{n})}$ that's closer than $\frac{ϵ}{2}$ to $y_{i}$ , and then since it's only $\frac{ϵ}{2}$ away from $y_{n}$ , we'd have that the distance from $y_{n}$ to $C_{K (x_{n})}$ is below $\frac{ϵ}{2}$ , an impossibility).

Because the distance from $y_{i}$ to any point in $C_{K (x_{n})}$ is above $\frac{ϵ}{2}$ , then

$| K (x_{n}) (f_{i}) - K (x) (f_{i}) | = | {inf}_{y \in C_{K (x_{n})}} f_{i} (y) - {inf}_{y \in C_{K (x)}} f_{i} (y) | = | 1 - 0 | = 1$

This is because $y_{i} \in C_{K (x)}$ and attains a value of 0 according to $f_{i}$ , while $C_{K (x_{n})}$ stays away from $y_{i}$ and all its points must have a value of 1. This situation happens infinitely often, which leads to a contradiction with

${lim}_{n \to \infty} {sup}_{i} | K (x_{n}) (f_{i}) - K (x) (f_{i}) | = 0$

Because infinitely often, one of these $f_{i}$ has very different values, so the sequence is 1 infinitely often and can't limit to 0.

So, we've ruled out that there is an $ϵ$ where, infinitely often, there is a point in $C_{K (x_{n})}$ that is $ϵ$ away from $C_{K (x)}$ . And we've ruled out that there is an $ϵ$ where, infinitely often, there is a point in $C_{K (x)}$ that is $ϵ$ away from $C_{K (x_{n})}$ . Fixing any $ϵ$ , in the tail of the sequence, $C_{K (x)}$ and $C_{K (x_{n})}$ are $ϵ$ distance or closer in Hausdorff distance because you can't find points in either set which are far away from the other set. So, $C_{K (x_{n})}$ limits to $C_{K (x)}$ in Hausdorff-distance when $x_{n}$ limits to $x$ , and we know that $x \mapsto C_{K (x)}$ is a continuous function $X \to K (Y)$ .

This lets us show that the set

$⋃_{x \in C_{h}} ({x} \times C_{K (x)})$

is closed, because if $x_{n}$ limits to $x$ and $y_{n} \in C_{K (x_{n})}$ and $y_{n}$ limits to $y$ , we have that $y \in C_{K (x)}$ because $C_{K (x_{n})}$ limits to $C_{K (x)}$ in Hausdorff distance, so we've got closed graph.

Also, by invoking Lemma 3, we know that

$⋃_{x \in C_{h}} C_{K (x)}$

is compact.

Time to wrap this all up. We know that $⋃_{x \in C_{h}} {x} \times C_{K (x)}$ is closed in $X \times Y$ from our Hausdorff limit argument. This set is also a subset of:

$C_{h} \times ⋃_{x \in C_{h}} C_{K (x)}$

Which is a product of two sets known to be compact, and is compact. It's a closed subset of a compact set, so it's compact. Therefore,

$⋃_{x \in C_{h}} {x} \times C_{K (x)}$

is a compact set, and from way back,

$(h ⋉ K) (f) = {inf}_{(x, y) \in ⋃_{x \in C_{h}} ({x} \times C_{K (x)})} f (x, y)$

And we've shown that set is compact, so $h ⋉ K$ where $h$ and all the $K (x)$ are sharp can be written as minimizing over a compact set, so $h ⋉ K$ is sharp. Thus, semidirect product preserves all the nice properties, and we're finally done with this proof.

Proposition 20: If all the $K (x)$ are C-additive, then $p r_{X *} (h ⋉ K) = h$ .

$p r_{X *} (h ⋉ K) (f) = (h ⋉ K) (f \circ p r_{X}) = h (λ x . K (x) (λ y . f (p r_{X} (x, y))))$

$= h (λ x . K (x) (λ y . f (x))) = h (λ x . f (x)) = h (f)$

This is because, since $f (x)$ doesn't depend on $y$ , it acts as a constant inside $K (x)$ and C-additivity lets us pull it out.

Proposition 21: If $K_{0}, K_{1}, K_{2} . . .$ are a sequence of infrakernels of type $K_{n} : \prod_{i = 0}^{i = n} X_{i} i k \to X_{n + 1}$ , and $h$ is an infradistribution over $X_{0}$ , then $(. . . ((h ⋉ K_{0}) ⋉ K_{1}) . . . ⋉ K_{m})$ can be rewritten as $h ⋉ K_{: m}$ where $K_{: n}$ is an infrakernel of type $X_{0} i k \to \prod_{i = 1}^{i = n + 1} X_{i}$ , recursively defined as $K_{: 0} := K_{0}$ and $K_{: n + 1} (x_{0}) := K_{: n} (x_{0}) ⋉ (λ x_{1 : n + 1} . K_{n + 1} (x_{0}, x_{1 : n + 1}))$

So, for our inductive definition,

$K_{: 0} (x_{0}) := K_{0} (x_{0})$

$K_{: n + 1} (x_{0}) := K_{: n} (x_{0}) ⋉ (λ x_{1 : n + 1} . K_{n + 1} (x_{0}, x_{1 : n + 1})$

Our task is to show that these are all infrakernels, by induction, and that for any infradistribution $h$ ,

$(. . . ((h ⋉ K_{0}) ⋉ K_{1}) . . . ⋉ K_{n}) = h ⋉ K_{: n}$

For the base case, we observe that $K_{: 0}$ is an infrakernel because it equals $K_{0}$ , which is an infrakernel, and that $h ⋉ K_{0} = h ⋉ K_{: 0}$

Time for the induction step. We'll assume that $K_{: n}$ is an infrakernel, and show that $K_{: n + 1}$ is. Further, we need to show that $h ⋉ K_{: n + 1} = (h ⋉ K_{: n}) ⋉ K_{n + 1}$ . This will show the result.

Our first requirement is showing that for all $x_{0}$ , $K_{: n + 1} (x_{0})$ is an infradistribution.

$K_{: n + 1} (x_{0}) := K_{: n} (x_{0}) ⋉ (λ x_{1 : n + 1} . K_{n + 1} (x_{0}, x_{1 : n + 1}))$

By our induction assumption, $K_{: n} (x_{0})$ is an infradistribution as $K_{: n}$ is an infrakernel. Further, $λ x_{1 : n + 1} . K_{n + 1} (x_{0}, x_{1 : n + 1})$ is an infrakernel because $K_{n + 1}$ is and we're just restricting it to a subset of its domain, so it keeps being an infrakernel. And we know from earlier that the semidirect product of an infradistribution and an infrakernel is an infradistribution. So that's taken care of.

Now, we must show a common Lipschitz constant, pointwise function convergence, and compact-shared compact almost-support for $K_{: n + 1}$ to certify that it's an infrakernel.

Starting with common Lipschitz constant, we can just note that, in our proof of Proposition 19, we saw that the Lipschitz constant of the semidirect product was upper-bounded by the product of the Lipschitz constants of the starting infradistributions and the kernel. Assuming that $K_{: n}$ is an infradistribution, we have that the Lipschitz constant of any $K_{: n} (x_{0})$ is upper-bounded by some $λ_{: n}^{⊙}$ Lipschitz constant. Also, the Lipschitz constant of $K_{n + 1} (x_{0}, x_{1 : n + 1})$ is upper-bounded by some $λ_{n + 1}^{⊙}$ Lipschitz constant. Thus, $λ_{: n}^{⊙} λ_{n + 1}^{⊙}$ is an upper-bound on the Lipschitz constant of any

$K_{: n} (x_{0}) ⋉ (λ x_{1 : n + 1} . K_{n + 1} (x_{0}, x_{1 : n + 1}))$

infradistribution, which is exactly $K_{: n + 1} (x_{0})$ , witnessing that $K_{: n + 1}$ has a uniform upper bound on its Lipschitz constants.

Time to move onto the second one, compact-shared compact almost-support.

For this one, we're trying to prove:

$\forall C^{X_{0}}, ϵ \exists C_{ϵ}^{\prod_{i = 1}^{i = n + 2} X_{i}} \subseteq \prod_{i = 1}^{i = n + 2} X_{i} \forall x_{0} \in C^{X_{0}}, f, f^{'} :$

$f_{↓ C_{ϵ}^{\prod_{i = 1}^{i = n + 2} X_{i}}} = f_{↓ C_{ϵ}^{\prod_{i = 1}^{i = n + 2} X_{i}}}^{'} \to | K_{: n + 1} (x_{0}) (f) - K_{: n + 1} (x_{0}) (f^{'}) | \leq ϵ d (f, f^{'})$

This is the sentence that says that $K_{: n + 1}$ has compact-shared compact almost-support. $f$ and $f^{'}$ have type signature $\prod_{i = 1}^{i = n + 2} X_{i} \to R$ .

Now, this is going to be quite complicated, so pay close attention. Fix an arbitrary compact $C^{X_{0}} \subseteq X_{0}$ , and an arbitrary $ϵ$ . Let $λ_{: n}^{⊙}$ be the Lipschitz constant for the infrakernel $K_{: n}$ , and $λ_{n + 1}^{⊙}$ be the Lipschitz constant for the infrakernel $K_{n + 1}$ .

Due to compact-shared compact-almost-support for $K_{: n}$ which exists by our induction assumption, your set $C^{X_{0}}$ induces a compact $\frac{ϵ}{2 λ_{n + 1}^{⊙}}$ -almost-support for the family of infradistributions $K_{: n} (x_{0})$ where $x_{0} \in C^{X_{0}}$ . Call said almost-support $C_{\frac{ϵ}{2 λ_{n + 1}^{⊙}}}^{\prod_{i = 1}^{i = n + 1} X_{i}}$ .

Further, due to compact-shared compact-almost-support for $K_{n + 1}$ , the set

$C^{X_{0}} \times C_{\frac{ϵ}{2 λ_{n + 1}^{⊙}}}^{\prod_{i = 1}^{i = n + 1} X_{i}}$

induces a compact $\frac{ϵ}{2 λ_{: n}^{⊙}}$ -almost-support for the family of infradistributions $K_{n + 1} (x_{0}, x_{1 : n + 1})$ where $(x_{0}, x_{1 : n + 1}) \in C^{X_{0}} \times C_{\frac{ϵ}{2 λ_{n + 1}^{⊙}}}^{\prod_{i = 1}^{i = n + 1} X_{i}}$

Call said almost-support $C_{\frac{ϵ}{2 λ_{: n}^{⊙}}}^{X_{n + 2}}$

And now let your shared $ϵ$ -almost-support for $K_{: n + 1} (x_{0})$ where $x_{0} \in C^{X_{0}}$ be:

$C_{\frac{ϵ}{2 λ_{n + 1}^{⊙}}}^{\prod_{i = 1}^{i = n + 1} X_{i}} \times C_{\frac{ϵ}{2 λ_{: n}^{⊙}}}^{X_{n + 2}}$

We must show that said set is indeed a shared $ϵ$ -almost-support for $K_{: n + 1} (x_{0})$ where $x_{0} \in C^{X_{0}}$ . So, let $f$ and $f^{'}$ agree on said set. Then, we have:

$| K_{: n + 1} (x_{0}) (f) - K_{: n + 1} (x_{0}) (f^{'}) |$

$= | (K_{: n} (x_{0}) ⋉ (λ x_{1 : n + 1} . K_{n + 1} (x_{0}, x_{1 : n + 1}))) (f) - (K_{: n} (x_{0}) ⋉ (λ x_{1 : n + 1} . K_{n + 1} (x_{0}, x_{1 : n + 1}))) (f^{'}) |$

$= | K_{: n} (x_{0}) (λ x_{1 : n + 1} . K_{n + 1} (x_{0}, x_{1 : n + 1}) (λ x_{n + 2} . f (x_{1 : n + 1}, x_{n + 2})))$

$- K_{: n} (x_{0}) (λ x_{1 : n + 1} . K_{n + 1} (x_{0}, x_{1 : n + 1}) (λ x_{n + 2} . f^{'} (x_{1 : n + 1}, x_{n + 2}))) |$

This is just unpacking the definition of the iterated semidirect product, no issues here. Now, we use Lemma 2 and the fact that $C_{\frac{ϵ}{2 λ_{n + 1}^{⊙}}}^{\prod_{i = 1}^{i = n + 1} X_{i}}$ is a $\frac{ϵ}{2 λ_{n + 1}^{⊙}}$ -almost-support for $K_{: n} (x_{0})$ when $x_{0} \in C^{X_{0}}$ , to get:

$\leq λ_{: n}^{⊙} {sup}_{x_{1 : n + 1} \in C_{\frac{ϵ}{2 λ_{n + 1}^{⊙}}}^{\prod_{i = 1}^{i = n + 1} X_{i}}} | K_{n + 1} (x_{0}, x_{1 : n + 1}) (λ x_{n + 2} . f (x_{1 : n + 1}, x_{n + 2}))$

$- K_{n + 1} (x_{0}, x_{1 : n + 1}) (λ x_{n + 2} . f^{'} (x_{1 : n + 1}, x_{n + 2})) |$

$+ \frac{ϵ}{2 λ_{n + 1}^{⊙}} {sup}_{x_{1 : n + 1}} | K_{n + 1} (x_{0}, x_{1 : n + 1}) (λ x_{n + 2} . f (x_{1 : n + 1}, x_{n + 2}))$

$- K_{n + 1} (x_{0}, x_{1 : n + 1}) (λ x_{n + 2} . f^{'} (x_{1 : n + 1}, x_{n + 2})) |$

Ok, this is a mess. Let's try to unpack

${sup}_{x_{1 : n + 1} \in C_{\frac{ϵ}{2 λ_{n + 1}^{⊙}}}^{\prod_{i = 1}^{i = n + 1} X_{i}}} | K_{n + 1} (x_{0}, x_{1 : n + 1}) (λ x_{n + 2} . f (x_{1 : n + 1}, x_{n + 2}))$

$- K_{n + 1} (x_{0}, x_{1 : n + 1}) (λ x_{n + 2} . f^{'} (x_{1 : n + 1}, x_{n + 2})) |$

first. What we can do is use that, regardless of what is picked in the supremum, we have:

$(x_{0}, x_{1 : n + 1}) \in C^{X_{0}} \times C_{\frac{ϵ}{2 λ_{n + 1}^{⊙}}}^{\prod_{i = 1}^{i = n + 1} X_{i}}$

So this means that

$C_{\frac{ϵ}{2 λ_{: n}^{⊙}}}^{X_{n + 2}}$

is a $\frac{ϵ}{2 λ_{: n}^{⊙}}$ -almost-support for $K_{n + 1} (x_{0}, x_{1 : n + 1})$ . Further, because $f$ and $f^{'}$ are identical on

$C_{\frac{ϵ}{2 λ_{n + 1}^{⊙}}}^{\prod_{i = 1}^{i = n + 1} X_{i}} \times C_{\frac{ϵ}{2 λ_{: n}^{⊙}}}^{X_{n + 2}}$

and $x_{1 : n + 1}$ was being selected from the former of those, then the functions $λ x_{n + 2} . f (x_{1 : n + 1}, x_{n + 2})$ (and the same for $f^{'}$ ) agree on $C_{\frac{ϵ}{2 λ_{: n}^{⊙}}}^{X_{n + 2}}$ , the almost-support. So, the supremum is upper-bounded by

$\leq \frac{ϵ}{2 λ_{: n}^{⊙}} d (f, f^{'})$

Substituting this back in, we get:

$\leq λ_{: n}^{⊙} \frac{ϵ}{2 λ_{: n}^{⊙}} d (f, f^{'}) + \frac{ϵ}{2 λ_{n + 1}^{⊙}} {sup}_{x_{1 : n + 1}} | K_{n + 1} (x_{0}, x_{1 : n + 1}) (λ x_{n + 2} . f (x_{1 : n + 1}, x_{n + 2}))$

$- K_{n + 1} (x_{0}, x_{1 : n + 1}) (λ x_{n + 2} . f^{'} (x_{1 : n + 1}, x_{n + 2})) |$

Now let's try to unpack

${sup}_{x_{1 : n + 1}} | K_{n + 1} (x_{0}, x_{1 : n + 1}) (λ x_{n + 2} . f (x_{1 : n + 1}, x_{n + 2}))$

$- K_{n + 1} (x_{0}, x_{1 : n + 1}) (λ x_{n + 2} . f^{'} (x_{1 : n + 1}, x_{n + 2})) |$

We don't have much leverage on it, besides using the basic Lipschitz constant upper bound, so let's try that.

$\leq {sup}_{x_{1 : n + 1}} λ_{n + 1}^{⊙} d (λ x_{n + 2} . f (x_{1 : n + 1}, x_{n + 2}), λ x_{n + 2} . f^{'} (x_{1 : n + 1}, x_{n + 2}))$

$= λ_{n + 1}^{⊙} d (f, f^{'})$

And substituting this back in, we get:

$\leq λ_{: n}^{⊙} \frac{ϵ}{2 λ_{: n}^{⊙}} d (f, f^{'}) + \frac{ϵ}{2 λ_{n + 1}^{⊙}} λ_{n + 1}^{⊙} d (f, f^{'})$

$= ϵ d (f, f^{'})$

And so we've shown that the functions are only $ϵ$ times their distance apart, so the compact set we cooked up is indeed an $ϵ$ -almost-support for $K_{: n + 1} (x_{0})$ whenever $x_{0} \in C^{X_{0}}$ , and because $ϵ$ and $C^{X_{0}}$ was arbitrary, we have compact-shared compact-almost-support for $K_{: n + 1}$ .

Time to move onto the third one, pointwise convergence. If $x_{0, m}$ limits to $x_{0, \infty}$ , we want $K_{: n + 1} (x_{0, m}) (f)$ to limit to $K_{: n + 1} (x_{0, \infty}) (f)$ . As usual, we use $λ_{n + 1}^{⊙}$ for the Lipschitz constant of $K_{n + 1}$ and $λ_{: n}^{⊙}$ for the Lipschitz constant of $K_{: n}$ .

To begin with, fix an arbitrary $ϵ$ and bounded continuous function $f$ , and note that ${x_{0, m}}_{m \in N \cup {\infty}}$ is a compact subset of $X_{0}$ . Because $K_{: n} : X_{0} i k \to \prod_{i = 1}^{i = n + 1} X_{i}$ is assumed to be an infrakernel by induction, ${x_{0, m}}_{m \in N \cup {\infty}}$ acts as a compact set for it. So, by compact-shared compact-almost-support for $K_{: n}$ , we can find a compact set $C_{\frac{ϵ}{4 λ_{n + 1}^{⊙} | | f | |}}^{\prod_{i = 1}^{i = n + 1} X_{i}}$ which is a $\frac{ϵ}{4 λ_{n + 1}^{⊙} | | f | |}$ -almost-support for $K_{: n}$ .

Also, it is important to note that

$λ x_{: n + 1} . K_{n + 1} (x_{: n + 1}) (λ x_{n + 2} . f (x_{: n + 1}, x_{n + 2}))$

Is a continuous function (as it must be for semidirect products with $K_{n + 1}$ to have the functions on the inside be continuous). Accordingly, this means that the function:

$λ x_{0}, x_{1 : n + 1} . K_{n + 1} (x_{0}, x_{1 : n + 1}) (λ x_{n + 2} . f (x_{1 : n + 1}, x_{n + 2}))$

must be uniformly continuous when restricted to the set ${x_{0, m}}_{m \in N \cup {\infty}} \times C_{\frac{ϵ}{4 λ_{n + 1}^{⊙} | | f | |}}^{\prod_{i = 1}^{i = n + 1} X_{i}}$
And so, by uniform continuity, given any $ϵ$ , there is some $δ$ difference in inputs which gives rise to a $\frac{ϵ}{2 λ_{: n}^{⊙}}$ difference in output.

Now, here's what we'll be doing. We'll attempt to show the result that

$\forall ϵ \exists m^{*} \forall m \geq m^{*} : | K_{: n} (x_{0, m}) (λ x_{1 : n + 1} . K_{n + 1} (x_{0, m}, x_{1 : n + 1}) (λ x_{n + 2} . f (x_{1 : n + 1}, x_{n + 2})))$

$- K_{: n} (x_{0, m}) (λ x_{1 : n + 1} . K_{n + 1} (x_{0, \infty}, x_{1 : n + 1}) (λ x_{n + 2} . f (x_{1 : n + 1}, x_{n + 2}))) | \leq ϵ$

Straight off the bat, we can apply Lemma 2 to decompose this difference into "starting Lipschitz constant times the difference of the inner functions on the compact set of interest" and "level of almost-support times the difference of the two functions", yielding:

$| K_{: n} (x_{0, m}) (λ x_{1 : n + 1} . K_{n + 1} (x_{0, m}, x_{1 : n + 1}) (λ x_{n + 2} . f (x_{1 : n + 1}, x_{n + 2})))$

$- K_{: n} (x_{0, m}) (λ x_{1 : n + 1} . K_{n + 1} (x_{0, \infty}, x_{1 : n + 1}) (λ x_{n + 2} . f (x_{1 : n + 1}, x_{n + 2}))) |$

$\leq λ_{: n}^{⊙} {sup}_{x_{1 : n + 1} \in C_{\frac{ϵ}{4 λ_{n + 1}^{⊙} | | f | |}}^{\prod_{i = 1}^{i = n + 1} X_{i}}} | K_{n + 1} (x_{0, m}, x_{1 : n + 1}) (λ x_{n + 2} . f (x_{1 : n + 1}, x_{n + 2}))$

$- K_{n + 1} (x_{0, \infty}, x_{1 : n + 1}) (λ x_{n + 2} . f (x_{1 : n + 1}, x_{n + 2})) |$

$+ \frac{ϵ}{4 λ_{n + 1}^{⊙} | | f | |} {sup}_{x_{1 : n + 1}} | K_{n + 1} (x_{0, m}, x_{1 : n + 1}) (λ x_{n + 2} . f (x_{1 : n + 1}, x_{n + 2}))$

$- K_{n + 1} (x_{0, \infty}, x_{1 : n + 1}) (λ x_{n + 2} . f (x_{1 : n + 1}, x_{n + 2})) |$

Time to start breaking this down. First, to break down

${sup}_{x_{1 : n + 1}} | K_{n + 1} (x_{0, m}, x_{1 : n + 1}) (λ x_{n + 2} . f (x_{1 : n + 1}, x_{n + 2}))$

$- K_{n + 1} (x_{0, \infty}, x_{1 : n + 1}) (λ x_{n + 2} . f (x_{1 : n + 1}, x_{n + 2})) |$

we can realize that the maximum value of one of these would be $λ_{n + 1}^{⊙} | | f | |$ , and the minimum possible value of one of these is $- λ_{n + 1}^{⊙} | | f | |$ , from Lipschitzness of $K_{n + 1}$ , producing an upper bound of:

$\leq 2 λ_{n + 1}^{⊙} | | f | |$

Substituting this back in, we have:

$- K_{n + 1} (x_{0, \infty}, x_{1 : n + 1}) (λ x_{n + 2} . f (x_{1 : n + 1}, x_{n + 2})) | + \frac{ϵ}{4 λ_{n + 1}^{⊙} | | f | |} 2 λ_{n + 1}^{⊙} | | f | |$

And now, we can use the fact that there is always some $δ$ difference in inputs which gives rise to a $\frac{ϵ}{2 λ_{: n}^{⊙}}$ difference in output of the function

$λ x_{0}, x_{1 : n + 1} . K_{n + 1} (x_{0}, x_{1 : n + 1}) (λ x_{n + 2} . f (x_{1 : n + 1}, x_{n + 2}))$

when restricted to the set ${x_{0, m}}_{m \in N \cup {\infty}} \times C_{\frac{ϵ}{4 λ_{n + 1}^{⊙} | | f | |}}^{\prod_{i = 1}^{i = n + 1} X_{i}}$
in order to find some $m^{*}$ where all future $m$ have $x_{0, m}$ being within $δ$ of $x_{0, \infty}$ .

This tiny difference means that the values

$K_{n + 1} (x_{0, m}, x_{1 : n + 1}) (λ x_{n + 2} . f (x_{1 : n + 1}, x_{n + 2}))$

and

$K_{n + 1} (x_{0, \infty}, x_{1 : n + 1}) (λ x_{n + 2} . f (x_{1 : n + 1}, x_{n + 2}))$

will only differ by $\frac{ϵ}{2 λ_{: n}^{⊙}}$ for all $x_{1 : n + 1}$ which lie in

$C_{\frac{ϵ}{4 λ_{n + 1}^{⊙} | | f | |}}^{\prod_{i = 1}^{i = n + 1} X_{i}}$

Therefore, we have that for all $m$ past $m^{*}$ ,

${sup}_{x_{1 : n + 1} \in C_{\frac{ϵ}{4 λ_{n + 1}^{⊙} | | f | |}}^{\prod_{i = 1}^{i = n + 1} X_{i}}} | K_{n + 1} (x_{0, m}, x_{1 : n + 1}) (λ x_{n + 2} . f (x_{1 : n + 1}, x_{n + 2}))$

$- K_{n + 1} (x_{0, \infty}, x_{1 : n + 1}) (λ x_{n + 2} . f (x_{1 : n + 1}, x_{n + 2})) | \leq \frac{ϵ}{2 λ_{: n}^{⊙}}$

And substituting this back in, we have:

$\leq λ_{: n}^{⊙} \frac{ϵ}{2 λ_{: n}^{⊙}} + \frac{ϵ}{4 λ_{n + 1}^{⊙} | | f | |} 2 λ_{n + 1}^{⊙} | | f | | = ϵ$

And $ϵ$ was arbitrary. Therefore we have our desired result that, regardless of bounded continuous function $f$ ,

$\forall ϵ \exists m^{*} \forall m \geq m^{*} : | K_{: n} (x_{0, m}) (λ x_{1 : n + 1} . K_{n + 1} (x_{0, m}, x_{1 : n + 1}) (λ x_{n + 2} . f (x_{1 : n + 1}, x_{n + 2})))$

$- K_{: n} (x_{0, m}) (λ x_{1 : n + 1} . K_{n + 1} (x_{0, \infty}, x_{1 : n + 1}) (λ x_{n + 2} . f (x_{1 : n + 1}, x_{n + 2}))) | \leq ϵ$

Therefore,

${lim}_{m \to \infty} | K_{: n} (x_{0, m}) (λ x_{1 : n + 1} . K_{n + 1} (x_{0, m}, x_{1 : n + 1}) (λ x_{n + 2} . f (x_{1 : n + 1}, x_{n + 2})))$

$- K_{: n} (x_{0, m}) (λ x_{1 : n + 1} . K_{n + 1} (x_{0, \infty}, x_{1 : n + 1}) (λ x_{n + 2} . f (x_{1 : n + 1}, x_{n + 2}))) | = 0$

These two things limit increasingly close to each other. Further,

${lim}_{m \to \infty} K_{: n} (x_{0, m}) (λ x_{1 : n + 1} . K_{n + 1} (x_{0, \infty}, x_{1 : n + 1}) (λ x_{n + 2} . f (x_{1 : n + 1}, x_{n + 2})))$

$= K_{: n} (x_{0, \infty}) (λ x_{1 : n + 1} . K_{n + 1} (x_{0, \infty}, x_{1 : n + 1}) (λ x_{n + 2} . f (x_{1 : n + 1}, x_{n + 2})))$

By pointwise convergence for $K_{: n}$ which is an infrakernel by our induction assumption. Putting these two parts together, we have:

${lim}_{m \to \infty} K_{: n} (x_{0, m}) (λ x_{1 : n + 1} . K_{n + 1} (x_{0, m}, x_{1 : n + 1}) (λ x_{n + 2} . f (x_{1 : n + 1}, x_{n + 2})))$

$= K_{: n} (x_{0, \infty}) (λ x_{1 : n + 1} . K_{n + 1} (x_{0, \infty}, x_{1 : n + 1}) (λ x_{n + 2} . f (x_{1 : n + 1}, x_{n + 2})))$
so

${lim}_{m \to \infty} (K_{: n} (x_{0, m}) ⋉ (λ x_{1 : n + 1} . K_{n + 1} (x_{0, m}, x_{1 : n + 1}))) (f)$

$= (K_{: n} (x_{0, \infty}) ⋉ (λ x_{1 : n + 1} . K_{n + 1} (x_{0, \infty}, x_{1 : n + 1}))) (f)$

${lim}_{m \to \infty} K_{: n + 1} (x_{0, m}) (f) = K_{: n + 1} (x_{0, \infty}) (f)$

And we're done, we showed pointwise convergence for $K_{: n + 1}$ which is the last condition necessary to show it's an infrakernel, and the induction proof goes through to show that all the $K_{: n}$ are infrakernels.

Now all that's left is to show that

$h ⋉ K_{: n + 1} = (h ⋉ K_{: n}) ⋉ K_{n + 1}$

using induction, we have the base case set up. We can go:

$(h ⋉ K_{: n + 1}) (f)$

$= h (λ x_{0} . K_{: n + 1} (x_{0}) (λ x_{1 : n + 2} . f (x_{0}, x_{1 : n + 2})))$

$= h (λ x_{0} . (K_{: n} (x_{0}) ⋉ (λ x_{1 : n + 1} . K_{n + 1} (x_{0}, x_{1 : n + 1}))) (λ x_{1 : n + 2} . f (x_{0}, x_{1 : n + 2})))$

$= h (λ x_{0} . K_{: n} (x_{0}) (λ x_{1 : n + 1} . K_{n + 1} (x_{0}, x_{1 : n + 1}) (λ x_{n + 2} . f (x_{0}, x_{1 : n + 1}, x_{n + 2}))))$

$= (h ⋉ K_{: n}) (λ x_{: n + 1} . K_{n + 1} (x_{: n + 1}) (λ x_{n + 2} . f (x_{: n + 1}, x_{n + 2})))$

$= ((h ⋉ K_{: n}) ⋉ K_{n + 1}) (f)$

And we're done. Because

$h ⋉ K_{: n + 1} = (h ⋉ K_{: n}) ⋉ K_{n + 1}$

and we know that $h ⋉ K_{: 0} = h ⋉ K_{0}$ , this means that

$\forall m : (. . . ((h ⋉ K_{0}) ⋉ K_{1}) . . . ⋉ K_{m}) = h ⋉ K_{: m}$

Proposition 22: $K_{: \infty}$ is an infrakernel (C-additive, specifically) if all the $K_{n}$ are C-additive infrakernels. It is unchanged by altering the $C_{i}$ sequence of compact sets. In addition, if all the $K_{n}$ are homogenous/cohomogenous/crisp/sharp, then $K_{: \infty}$ will be so as well.

So, $K_{: \infty} : X_{0} i k \to \prod_{i = 1}^{\infty} X_{i}$ is defined as: Fixing an arbitrary sequence of compact sets $C_{i} \in X_{i}$ ,
$K_{: \infty} (x_{0}) (f) := {lim}_{n \to \infty} K_{: n} (x_{0}) (λ x_{1 : n + 1} {inf}_{x_{n + 2 : \infty} \in \prod_{i = n + 2}^{\infty} C_{i}} f (x_{1 : n + 1}, x_{n + 2 : \infty}))$
Is it an infrakernel?

This is going to suck unbelievably much, we're gonna need a ton of results. The game plan is:

Part 1: Show that the functions you're feeding into those infrakernels are guaranteed to be continuous, to make some progress towards showing that $K_{: \infty}$ is well-defined.

Part 2: Show that all the $K_{: n}$ are 1-Lipschitz, and also preserve all nice properties we'd want if all the $K_{n}$ do (homogenity, cohomogenity, C-additity, crispness, sharpness).

Part 3: Show that if a function only depends on the first n coordinates of the input, then all the $K_{: n + m}$ start agreeing on the expectation value of the function.

Part 4: Give a general procedure for taking a compact subset of the space $X_{0}$ and making a compact subset of the space $\prod_{i = 1}^{\infty} X_{i}$ with nice properties related to compact almost-support, that preserves its nice properties when projected down to any finite stage.

Part 5: Use parts 2, 3, 4, and a complicated chain of reasoning to get a result which implies that it doesn't matter which $C_{i}$ sequence you pick, the limit will exist and be same for all of them, so $K_{: \infty}$ actually exists and is well-defined.

Part 6: Using parts 2 and 5, clean up the normalization, monotonicity, concavity, and C-additivity properties of $K_{: \infty}$ . Showing that all the $K_{: \infty} (x_{0})$ are C-additive trivially nets the bounded Lipschitz constant property to show that $K_{: \infty}$ is an infrakernel and $K_{: \infty} (x_{0})$ is an infradistribution.

Part 7: Use our trick from Part 4 and our freedom of picking our compact set sequence from Part 5 to show compact-shared compact almost-support for $K_{: \infty}$ , netting us the second infrakernel property, and the compact almost-support property for all the individual components of kernel, verifying the last condition we need to conclude that $K_{: \infty} (x_{0})$ is an infradistribution.

Part 8: We recap one of the arguments for part 5, and it lets us get uniform convergence for a certain limit on any compact set, which is a critical lemma for Part 9.

Part 9: We use our result from Part 8 to invoke the Moore-Osgood theorem in order to show pointwise convergence for $K_{: \infty}$ , wrapping up the last condition for it to be an infrakernel.

Part 10: Show that if all the $K_{: n}$ have some nice property, then the limit $K_{: \infty}$ inherits it too.

The proofs will proceed in a strange way to keep track of all the moving parts in places. We'll first present the thing we're trying to prove, and repeatedly go "we could prove it if we could prove this other thing", and keep chaining back until we get something that's easy to show.

Proof Part 1: Our desired result is whether the function $λ x_{1 : n + 1} . {inf}_{x_{n + 2 : \infty} \in \prod_{i = n + 2}^{\infty} C_{i}} f (x_{1 : n + 1}, x_{n + 2 : \infty})$
is continuous. So, letting $x_{1 : n + 1}^{m}$ limit to $x_{1 : n + 1}^{\infty}$ , our task is to show that:

${lim}_{m \to \infty} {inf}_{x_{n + 2 : \infty} \in \prod_{i = n + 2}^{\infty} C_{i}} f (x_{1 : n + 1}^{m}, x_{n + 2 : \infty}) = {inf}_{x_{n + 2 : \infty} \in \prod_{i = n + 2}^{\infty} C_{i}} f (x_{1 : n + 1}^{\infty}, x_{n + 2 : \infty})$

Now, what we can do is consider the compact subset of $\prod_{i = 1}^{i = \infty} X_{i}$ to be
${x_{1 : n + 1}^{m}}_{m \in N \cup {\infty}} \times \prod_{i = n + 2}^{\infty} C_{i}$

And then $f$ must be uniformly continuous on it, so given any $ϵ$ , there is some $δ$ where points only $δ$ away lead to only an $ϵ$ differ in value. You can consider $m$ to be big enough to guarantee that all future values of $x_{1 : n + 1}^{m}$ are within $δ$ of $x_{1 : n + 1}^{\infty}$ , and then this gets that the function values can only differ by $ϵ$ between $(x_{1 : n + 1}^{m}, x_{n + 2 : \infty})$ and $(x_{1 : n + 1}^{\infty}, x_{n + 2 : \infty})$ if $x_{n + 2 : \infty} \in \prod_{i = n + 2}^{\infty} C_{i}$ , which it is. This ensures that the worst-case function values are only $ϵ$ apart. This works for all $ϵ$ , showing

And so, all the functions we're feeding into the $K_{: n} (x_{0})$ are continuous.

Proof Part 2: Desired result is "if all the $K_{n}$ have a nice property, then all the $K_{: n}$ have it too".

This can be simply addressed by noting that, for the base case, because $K_{: 0} = K_{0}$ and we're assuming all the $K_{n}$ have (C-additivity/cohomogenity/homogenity/crispness/sharpness), $K_{: 0}$ trivially fulfills it.

And for the induction step, if we assume that $K_{: n}$ is 1-Lipschitz, note that:

$K_{: n + 1} (x_{0}) = K_{: n} (x_{0}) ⋉ (λ x_{1 : n + 1} . K_{n + 1} (x_{0}, x_{1 : n + 1}))$

And, by our results on semidirect products preserving nice properties, if $K_{: n} (x_{0})$ has the nice property (by induction assumption) and $K_{n + 1}$ does, then we get that $K_{: n + 1} (x_{0})$ preserves the same property, and it holds all the way up the $K_{: n}$ .
And we can move on to Part 3.

Part 3: Showing that, if we go far enough out in the $K_{: n}$ , the value assigned to functions which only depend on finitely many inputs stabilizes. The result that we'd like to show at this point is:

$\forall x_{0} \in X_{0}, n, m \in N, f \in C_{B} (\prod_{i = 1}^{i = n + 1} X_{i}) : K_{: n + m} (x_{0}) (f) = K_{: n} (x_{0}) (f)$

Admittedly, f is not of the proper type signature to be evaluated by $K_{: n + m} (x_{0})$ , but we're abusing notation so that we can feed it in anyways and it just ignores all the coordinates it doesn't need. Accordingly, fix an arbitrary $x_{0}, n, f,$ and our proof target will now be:

$\forall m \in N : K_{: n + m} (x_{0}) (f) = K_{: n + m + 1} (x_{0}) (f)$

Proving this would let you apply induction, because we have a base case where $K_{: n + 0} (x_{0}) (f) = K_{: n} (x_{0}) (f)$ . Let $m$ be arbitrary. Then, we can go:

$K_{: n + m + 1} (x_{0}) (f) = K_{: n + m} (x_{0}) (λ x_{1 : n + m + 1} . K_{n + m + 1} (x_{0}, x_{1 : n + m + 1}) (λ x_{n + m + 2} . f (x_{1 : n + 1})))$

And then, since the function doesn't depend on the choice of $x_{n + m + 2}$ , it's a constant and C-additivity of $K_{n + m + 1}$ lets us pull it out, yielding

$= K_{: n + m} (x_{0}) (λ x_{1 : n + m + 1} . f (x_{1 : n + 1})) = K_{: n + m} (x_{0}) (f)$

And we're done.

Part 4: Our desired result here is:

$\forall C^{X_{0}} \subseteq X_{0}, ϵ > 0 : \exists C_{1 : \infty} [C^{X_{0}}, ϵ] \subseteq \prod_{i = 1}^{\infty} X_{i} : \forall n \in N, x_{0} \in C^{X_{0}}, f, f^{'} \in C_{B} (\prod_{i = 1}^{n + 1} X_{i}) :$

$f_{↓ p r_{1 : n + 1} (C_{1 : \infty} [C^{X_{0}}, ϵ])} = f_{↓ p r_{1 : n + 1} (C_{1 : \infty} [C^{X_{0}}, ϵ])}^{'}$

$\to | K_{: n} (x_{0}) (f) - K_{: n} (x_{0}) (f^{'}) | \leq ϵ (1 - \frac{1}{2^{n + 1}}) d (f, f^{'})$

This is a bit complicated. It's saying that if you pick any compact subset of $X_{0}$ , you can make a compact subset of $\prod_{i = 1}^{\infty} X_{i}$ where the projection of it to coordinates 1 through $n + 1$ acts as a compact $ϵ (1 - \frac{1}{2^{n + 1}})$ -almost-support for all the $K_{: n} (x_{0})$ infradistributions when $x_{0}$ lies in your compact subset of $X_{0}$ . Regardless of what $n$ is.

Accordingly, fix some $C^{X_{0}}$ and $ϵ$ . Now, we can recursively build up compact subsets of all the $X_{n}$ in the following way.

$C_{\frac{ϵ}{2^{n + 1}}}^{X_{n + 1}} \subseteq X_{n + 1}$ is a $\frac{ϵ}{2^{n + 1}}$ -almost-support for all the $K_{n} (x_{1 : n})$ where $x_{1 : n} \in C^{X_{0}} \times \prod_{i = 1}^{i = n} C_{\frac{ϵ}{2^{i}}}^{X_{i}}$ . So, basically, we're recursively building up compact subsets of $\prod_{i = 0}^{i = n} X_{i}$ by taking products of earlier compact subsets (with your base case being $C^{X_{0}}$ ), and then going "that's a compact subset of the input to $K_{n}$ , we must be able to find a compact subset of $X_{n + 1}$ that's a $\frac{ϵ}{2^{n + 1}}$ -almost-support for all the $K_{n} (x_{: n})$ where $x_{: n}$ lies in our compact subset of input, because of the compact-shared almost-support condition for all the $K_{n}$ " to go to the next compact set.

To establish some notation to make this a bit easier, let

$C_{i} [C^{X_{0}}, ϵ] := C_{\frac{ϵ}{2^{i + 1}}}^{X_{i}}$

(the i'th compact set in the sequence, defined with $C^{X_{0}}$ to start building your sequence), and let

$C_{1 : n} [C^{X_{0}}, ϵ] := \prod_{i = 1}^{i = n} C_{i} [C^{X_{0}}]$

(the product of compact sets 1 through $n$ , which is compact)

And let

$C_{1 : \infty} [C^{X_{0}}, ϵ] := \prod_{i = 1}^{\infty} C_{i} [C^{X_{0}}]$

This is the product of all the compact sets, and is compact.

Note the dependence of these on the starting compact sets. Notice that the projection of $C_{1 : \infty} [C^{X_{0}}, ϵ]$ to coordinates 1 through $n$ is exactly $C_{1 : n} [C^{X_{0}}, ϵ]$ .

Now that this is established, our proof target is (using our new notation):

$\forall n \in N, x_{0} \in C^{X_{0}}, f, f^{'} \in C_{B} (\prod_{i = 1}^{n + 1} X_{i}) :$

$f_{↓ C_{1 : n + 1} [C^{X_{0}}, ϵ])} = f_{↓ C_{1 : n + 1} [C^{X_{0}}, ϵ])}^{'} \to | K_{: n} (x_{0}) (f) - K_{: n} (x_{0}) (f^{'}) | \leq ϵ (1 - \frac{1}{2^{n + 1}}) d (f, f^{'})$

This structure naturally suggests an induction proof, so for the base case, let our number be 0. Our proof target then turns into:

$\forall x_{0} \in C^{X_{0}}, f, f^{'} \in C_{B} (X_{1}) :$

$f_{↓ C_{1} [C^{X_{0}}, ϵ])} = f_{↓ C_{1} [C^{X_{0}}, ϵ])}^{'} \to | K_{: 0} (x_{0}) (f) - K_{: 0} (x_{0}) (f^{'}) | \leq ϵ (1 - \frac{1}{2}) d (f, f^{'})$

Using that $K_{: 0} = K_{0}$ and that $C_{1} [C^{X_{0}}, ϵ] = C_{\frac{ϵ}{2}}^{X_{1}}$ and $ϵ (1 - \frac{1}{2}) = \frac{ϵ}{2}$ our proof target is now:

$\forall x_{0} \in C^{X_{0}}, f, f^{'} \in C_{B} (X_{1}) :$

$f_{↓ C_{\frac{ϵ}{2}}^{X_{1}}} = f_{↓ C_{\frac{ϵ}{2}}^{X_{1}}}^{'} \to | K_{0} (x_{0}) (f) - K_{0} (x_{0}) (f^{'}) | \leq \frac{ϵ}{2} d (f, f^{'})$

However, we constructed $C_{\frac{ϵ}{2}}^{X_{1}}$ to be a $\frac{ϵ}{2}$ -almost-support for all the $K_{0} (x_{0})$ where $x_{0} \in C^{X_{0}}$ , so this statement is just true, and we're done with our base case.

Now, for the induction step, our proof target is:

$\forall x_{0} \in C^{X_{0}}, f, f^{'} \in C_{B} (\prod_{i = 1}^{n + 1} X_{i}) : f_{↓ C_{1 : n + 1} [C^{X_{0}}, ϵ]} = f_{↓ C_{1 : n + 1} [C^{X_{0}}, ϵ]}^{'}$

$\to | K_{: n} (x_{0}) (f) - K_{: n} (x_{0}) (f^{'}) | \leq ϵ (1 - \frac{1}{2^{n + 1}}) d (f, f^{'})$

implies

$\forall x_{0} \in C^{X_{0}}, f, f^{'} \in C_{B} (\prod_{i = 1}^{n + 2} X_{i}) : f_{↓ C_{1 : n + 2} [C^{X_{0}}, ϵ]} = f_{↓ C_{1 : n + 2} [C^{X_{0}}, ϵ]}^{'}$

$\to | K_{: n + 1} (x_{0}) (f) - K_{: n + 1} (x_{0}) (f^{'}) | \leq ϵ (1 - \frac{1}{2^{n + 2}}) d (f, f^{'})$

Accordingly, assume

$\forall x_{0} \in C^{X_{0}}, f, f^{'} \in C_{B} (\prod_{i = 1}^{n + 1} X_{i}) : f_{↓ C_{1 : n + 1} [C^{X_{0}}, ϵ]} = f_{↓ C_{1 : n + 1} [C^{X_{0}}, ϵ]}^{'}$

$\to | K_{: n} (x_{0}) (f) - K_{: n} (x_{0}) (f^{'}) | \leq ϵ (1 - \frac{1}{2^{n + 1}}) d (f, f^{'})$

And our task is to prove

$\forall x_{0} \in C^{X_{0}}, f, f^{'} \in C_{B} (\prod_{i = 1}^{n + 2} X_{i}) : f_{↓ C_{1 : n + 2} [C^{X_{0}}, ϵ]} = f_{↓ C_{1 : n + 2} [C^{X_{0}}, ϵ]}^{'}$

$\to | K_{: n + 1} (x_{0}) (f) - K_{: n + 1} (x_{0}) (f^{'}) | \leq ϵ (1 - \frac{1}{2^{n + 2}}) d (f, f^{'})$

Therefore let $x_{0}, f, f^{'}$ be arbitrary, and remember that they have the indicated properties, and that $f, f^{'}$ agree with each other on the indicated set $C_{1 : n + 2} [C^{X_{0}}, ϵ]$ . Our proof target is now:

$| K_{: n + 1} (x_{0}) (f) - K_{: n + 1} (x_{0}) (f^{'}) | \leq ϵ (1 - \frac{1}{2^{n + 2}}) d (f, f^{'})$

Unpacking the definition of $K_{: n + 1}$ and rewriting the thing on the end, this is equivalent to (we now take this as the proof target)

$| K_{: n} (x_{0}) (λ x_{1 : n + 1} . K_{n + 1} (x_{0}, x_{1 : n + 1}) (λ x_{n + 2} . f (x_{1 : n + 1}, x_{n + 2})))$

$- K_{: n} (x_{0}) (λ x_{1 : n + 1} . K_{n + 1} (x_{0}, x_{1 : n + 1}) (λ x_{n + 2} . f^{'} (x_{1 : n + 1}, x_{n + 2}))) |$

$\leq ϵ (1 - \frac{1}{2^{n + 1}}) d (f, f^{'}) + \frac{ϵ}{2^{n + 2}} d (f, f^{'})$

We can apply the Lemma 2 decomposition, to split this into "level of support of compact set x distance of functions + distance of functions on compact set x lipschitz constant of infradistribution". So, theoretically, if we had the following two results:

$\forall x_{1 : n + 1} \in C_{1 : n + 1} [C^{X_{0}}, ϵ] : | K_{n + 1} (x_{0}, x_{1 : n + 1}) (λ x_{n + 2} . f (x_{1 : n + 1}, x_{n + 2}))$

$- K_{n + 1} (x_{0}, x_{1 : n + 1}) (λ x_{n + 2} . f^{'} (x_{1 : n + 1}, x_{n + 2})) | \leq \frac{ϵ}{2^{n + 2}} d (f, f^{'})$

and

$\forall x_{1 : n + 1} : | K_{n + 1} (x_{0}, x_{1 : n + 1}) (λ x_{n + 2} . f (x_{1 : n + 1}, x_{n + 2}))$

$- K_{n + 1} (x_{0}, x_{1 : n + 1}) (λ x_{n + 2} . f^{'} (x_{1 : n + 1}, x_{n + 2})) | \leq d (f, f^{'})$

then applying Lemma 2 would get us

$| K_{: n} (x_{0}) (λ x_{1 : n + 1} . K_{n + 1} (x_{0}, x_{1 : n + 1}) (λ x_{n + 2} . f (x_{1 : n + 1}, x_{n + 2})))$

$- K_{: n} (x_{0}) (λ x_{1 : n + 1} . K_{n + 1} (x_{0}, x_{1 : n + 1}) (λ x_{n + 2} . f^{'} (x_{1 : n + 1}, x_{n + 2}))) |$

$\leq ϵ (1 - \frac{1}{2^{n + 1}}$