Let G be a finite group of measure-preserving measurable maps on (Ω, 𝒜, 𝐏) and let 𝒜0:={A ∈𝒜: g(

Let G be a finite group of measure-preserving measurable...

Key Concepts

Measure Preserving Transformation

A measure preserving transformation is a mapping on a measurable space that leaves the measure unchanged; that is, for any measurable set, the measure of the set is equal to the measure of its image under the transformation. Such transformations are central in ergodic theory and ensure that probabilistic properties remain invariant under the action of the transformation.

Invariant Sigma-Algebra

An invariant sigma-algebra with respect to a group of transformations consists of all measurable sets that are unchanged by every transformation in the group. This concept allows one to focus on the portions of a probability space that remain constant under group actions, providing a framework for studying symmetry and invariance in measure theoretic contexts.

Conditional Expectation

Conditional expectation, in the context of a sigma-algebra, is a projection of a random variable onto the space of sigma-algebra-measurable functions. It provides the best approximation of the random variable in the sense of mean square error given the information captured by the sigma-algebra, and it has a fundamental role in probability theory and stochastic processes.

Averaging Operator over Group Actions

The averaging operator over a finite group applies to a function by taking the average of its images under the group's actions. This operator projects the function onto the subspace of functions that are invariant under the group, thereby linking group symmetries with the concept of conditional expectation with respect to the invariant sigma-algebra.

Transcript

00:01 In this question, we are given a random variable, which goes from omega script a to the real line of borel set r.

00:16 We say x is a random variable.

00:22 We define x as a random variable for any set which is contained in script b.

00:40 Then f inverse of b is contained in script a.

00:47 Right, this is a definition for a random variable.

00:50 Here, we want to define something else.

00:53 We define script f as a set a, such that a is equal to x inverse of b, where b is a set in script of b.

01:10 We want to show x is also a variable function from omega.

01:19 Now, it's not from a, it is from f to real line script b.

01:32 I mean, for any random variable x, we can shrink the sigma algebra.

01:39 I mean, we can define some new sigma algebra such that x is measurable on this smaller sigma algebra because it's easy to see we have this containing relationship.

01:56 F must be a sub sigma algebra for a.

02:00 Every random variable, we can change the domain or we can change our sigma algebra so that it is measurable on some smaller thing.

02:11 Now, we want to show x is measurable on f.

02:16 This is very straightforward because that's the reason for us to define something strange, something not so natural.

02:32 We want to use this not so natural definition to make x measurable on f.

02:41 Okay, let's do it.

02:43 We say to prove x is measurable, we need to pick any set b in script b and consider the pre -image of b under x.

03:00 By the definition, this is an element in f.

03:06 Okay, for any b because of our construction.

03:15 That means for any set b which is contained in script of b, the pre -image of b under x must be an element in our f.

03:28 This means x is a measurable function from the new sigma algebra to this guy.

03:52 Okay, this is the simple solution for this question.

03:56 But here, we actually need to prove something more because under all of our discussion about, we just assume f is a sigma algebra.

04:10 But we have not proved it yet...

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