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The exercise you shared involves random variables \( X \) and \( Y \), and the goal is to prove that \( Y_2 \), the difference between \( Y \) and its conditional expectation \( E[Y | X] \), is uncorrelated with every \( \sigma(X) \)-measurable random variable.

Here is a detailed step-by-step explanation of the solution:

### Problem Restatement

Let \( X \) and \( Y \) be integrable random variables on a probability space \( (\Omega, \mathcal{F}, \mathbb{P}) \). Define:
- \( Y = Y_1 + Y_2 \), where
  - \( Y_1 = E[Y | X] \), the conditional expectation of \( Y \) given \( X \), which is \( \sigma(X) \)-measurable,
  - \( Y_2 = Y - E[Y | X] = Y - Y_1 \), the difference between \( Y \) and its conditional expectation.

The task is to show that:
1. \( Y_2 \) and \( X \) are uncorrelated.
2. More generally, \( Y_2 \) is uncorrelated with every \( \sigma(X) \)-measurable random variable.

### Key Definitions and Setup

- **Uncorrelated** means that the covariance between two random variables \( Z_1 \) and \( Z_2 \) is zero, i.e., \( \text{Cov}(Z_1, Z_2) = E[(Z_1 - E[Z_1])(Z_2 - E[Z_2])] = 0 \).
- The conditional expectation \( E[Y | X] \) is \( \sigma(X) \)-measurable, which means it is a function of \( X \) and depends only on \( X \).

### Proof (More General Case)

We are asked to show that \( Y_2 \) is uncorrelated with every \( \sigma(X) \)-measurable random variable \( \xi \).

By definition, \( Y_2 = Y - E[Y | X] \). To show uncorrelation with \( \xi \), we need to show that:
\[
E[Y_2 \xi] = 0
\]

Now, proceed as follows:

1. **Expectation of the product**:  
   \[
   E[Y_2 \xi] = E\left[(Y - E[Y | X])\xi\right] = E[Y \xi] - E[E[Y | X] \xi]
   \]

2. **Simplification of terms**:  
   By the property of conditional expectation:
   \[
   E[E[Y | X] \xi] = E[\xi E[Y | X]]
   \]
   Since \( \xi \) is \( \sigma(X) \)-measurable, it can be factored out of the conditional expectation:
   \[
   E[\xi E[Y | X]] = E[\xi \xi] = E[\xi]
   \]
   Which simplifies it to zero.


This solves the

Let X and Y be integrable random variables on a probability space (Ω,F,P). Then Y = Y1 + Y2, where Y1 = E[Y|X] is σ(X)-measurable and Y2 = Y − E[Y|X]. Show that Y2 and X are uncorrelated. More generally, show that Y2 is uncorrelated with every σ(X)-measurable random variable.

Explore a detailed step-by-step solution to show that Y2, the difference between Y and E[Y | X], is uncorrelated with σ(X)-measurable random variables. This problem involves random variables, conditional expectation, and properties of uncorrelated variables, suitable for students in advanced probability theory courses.

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