This is a technical post researching infinite factor spaces

In a factor space , for a measurable , a -measurable index function is generating , if depends only on those arguments , where is , and . In this post, we show that there is an almost surely minimal such index function that we call the history of given .

We will reintroduce all the definitions from the finite case:

Let be finite. Let be a measure space with nullsets, i.e. is a set, is a -algebra and is a -ideal that admits a probability, i.e. there is with .

We define a product -ideal: Clearly, . We can extend this by induction to .

We construct . Furthermore, let .

Definition 1 (almost sure subset)

We will write .

Definition 2 (almost sure union with arbitrary index set)

If is a -ideal that admits a probability and is a family of measurable sets, then there exists an almost surely unique and minimal with i.e. whenever , we have . We set Furthermore, there is countable, s.t. .
Proof. We will construct .

To start, choose any , and let .

Let be defined and let . Choose with and set and .

We set and

  • A contains : Let , we have to show . Assume not, then , so for all . Therefore, we have . Now , which is a contradiction.

  • Minimality and uniqueness: Now let . Then clearly, . If is also minimal, then clearly

Definition 3 (almost sure intersection)

We set .

Definition 4 (feature)

We call a measurable function a feature. In the following let be features.

Definition 5 (index function)

We call a measurable an index function. We write . We identify with , where to allow for set operations such as .

Definition 6

For an index function , we set .

Definition 7

We define that is almost surely a function of by .

Let be all product distributions whose nullsets are .

Definition 8 (conditional generation)

Let be . We write , if . Note that

Lemma 9

For let be and .

Then . Proof. Let , .

'': Obviously, . Let , then there is a with

'': Obviously, . Let , then there is with

Lemma 10

Let and . Then .

Proof. Let . We first show . For , let where etc. Let . We have Now

Now since sets of the form generate and are -stable, we have that .

It remains to show . Since , we have . Since the same holds true for , we get and therefore .

We claim .

'': Trivial.
'': Let and . Then . Therefore, for a . We claim . We have , we have . Similarly, and therefore .

Lemma 11

For , let . Let . Then .
Proof. Let . Clearly, and .

We first show . Since , we have . Now . Clearly, is -stable and by Lemma 9 a generator of . Therefore, .

We now show . Let and

'" : Trivial.
'': From , we have Therefore, .

Clearly, is a -stable generator of and therefore . Now, since , we have and therefore .

Definition 12 (history)

is the a.s. smallest generating index function.

Theorem 13 A history exists and is a.s. unique.

Proof. Existence: Let . Define . Now clearly, for each , there are countable many , such that . Now , and therefore . Furthermore, by construction we have for any , that which implies .

Uniqueness: Let be minimal, then clearly, .

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