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This is a technical post researching infinite factor spaces

In a factor space $F = ⨉_{i \in I} F_{i}$ , for a measurable $Z : F \to Z (F)$ , a $Z$ -measurable index function $J : F \to {0, 1}^{I}$ is generating $X : F \to X (F)$ , if $X$ depends only on those arguments $(x_{i})$ , where $J (x)_{i}$ is $1$ , and $Z$ . In this post, we show that there is an almost surely minimal such index function that we call the history of $X$ given $Z$ .

We will reintroduce all the definitions from the finite case:

Let $I$ be finite. Let $(F_{i}, A_{i}, N_{i})$ be a measure space with nullsets, i.e. $F_{i}$ is a set, $A_{i}$ is a $σ$ -algebra and $N_{i}$ is a $σ$ -ideal that admits a probability, i.e. there is $P_{i}$ with $N_{i} = {A \in A : P_{i} (A) = 0}$ .

We define a product $σ$ -ideal: $N_{1} \otimes N_{2} = {A \in A : {x : A (x) \notin N_{2}} \in N_{1}} .$ Clearly, $N_{1} \otimes N_{2} = {A \in A : P_{1} \times P_{2} (A) = 0}$ . We can extend this by induction to $⨂_{i \in I} N_{i}$ .

We construct $(F, A, N) = (⨉_{i \in I} F, ⨂_{i \in I} A_{i}, ⨂_{i \in I} N_{i})$ . Furthermore, let $P = ⨉_{i \in I} P_{i}$ .

Definition 1 (almost sure subset)

We will write $A * \subseteq B :\Leftrightarrow A ∖ B \in N$ .

Definition 2 (almost sure union with arbitrary index set)

If $N$ is a $σ$ -ideal that admits a probability $P$ and ${(A_{j})}_{j \in J} \in A^{J}$ is a family of measurable sets, then there exists an almost surely unique and minimal $A \in A$ with $\forall j \in J : A_{j} * \subseteq A .$ i.e. whenever $\forall j \in J : A_{j} * \subseteq B$ , we have $A * \subseteq B$ . We set ${* ⋃}_{j \in J} A_{j} ≔ A .$ Furthermore, there is $J^{*} \subseteq J$ countable, s.t. ${⋃^{*}}_{j \in J} A_{j} = ⋃_{j \in J^{*}} A_{j}$ .
Proof. We will construct $J^{*}$ .

To start, choose any $j \in J$ , $J_{0} = {j}$ and let $B_{0} = A_{j}$ .

Let $B_{n}$ be defined and let $s_{n} = sup j \in J P (A_{j} ∖ B_{n})$ . Choose $j_{n} \in J$ with $P ({A_{j}}_{n} ∖ B_{n}) \geq s_{n} - \frac{1}{n}$ and set $J_{n + 1} = J_{n} \cup {j_{n}}$ and $B_{n + 1} = B_{n} \cup {A_{j}}_{n}$ .

We set $J^{*} = {j_{n} : n \in N}$ and $A = ⋃_{j \in J^{*}} A_{j} .$

A contains $A_{j}$ : Let $j \in J$ , we have to show $P (A_{j} ∖ A) = 0$ . Assume not, then $P (A_{j} ∖ B_{n}) ↓ P (A_{j} ∖ A) = p > 0$ , so $s_{n} \geq p$ for all $n$ . Therefore, we have $P ({A_{j}}_{n} ∖ B_{n}) \geq p - \frac{1}{n}$ . Now $1 \geq P (A) = P (⨃_{n \in N} {A_{j}}_{n} ∖ B_{n}) = \sum_{n \in N} P ({A_{j}}_{n} ∖ B_{n}) = \infty$ , which is a contradiction.
Minimality and uniqueness: Now let $B \in A : \forall j \in J : A_{j} * \subseteq B$ . Then clearly, $A = ⋃_{n \in N} {A_{j}}_{n} * \subseteq B$ . If $B$ is also minimal, then clearly $A * \subseteq B * \subseteq A \Leftrightarrow P (A ▲ B) = 0 \Leftrightarrow A a.s. = B .$

Definition 3 (almost sure intersection)

We set ${⋂^{*}}_{j \in J} A_{j} = {({⋃^{*}}_{j \in J} A_{j}^{c})}_{j \in J}^{c}$ .

Definition 4 (feature)

We call a measurable function $Z : F \to Z (F)$ a feature. In the following let $X, Y, Z$ be features.

Definition 5 (index function)

We call a measurable $J : F \to {0, 1}^{I}$ an index function. We write $J_{i} = π_{i} \circ J$ . We identify $J$ with $J^{'} : F \to P (I)$ , where $J^{'} (x) = {i \in I : J_{i} (x) = 1}$ to allow for set operations such as $J \cap K = m i n (J, K)$ .

Definition 6

For an index function $J$ , we set $π_{J} = {(1_{{J_{i} = 1}} π_{i})}_{i \in I}$ .

Definition 7

We define that $X$ is almost surely a function of $Y$ by $X ≲^{*} Y :\Leftrightarrow σ (X) \subseteq σ (Y, N)$ .

Let $▲^{*} (F, N)$ be all product distributions whose nullsets...

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