Measure Theory // 2

Characterize a conditional mean m(X) by the regional averaging identity E[Y1{X ∈ A}] = E[m(X)1{X ∈ A}] for every measurable set A.

Show that E[Y | X] minimizes mean squared prediction error over all square-integrable functions of X and derive the associated residual orthogonality condition.

Derive population linear regression as the best linear approximation to E[Y | X] and distinguish the linear predictor from the true conditional mean.

For integrable Y, construct the measure ν(A) = E[Y1{X ∈ A}] on the space of X-values and verify that ν is absolutely continuous with respect to the distribution P_X.

Apply the Radon-Nikodym theorem to identify dν/dP_X with a version of the conditional mean m(x).

Explain why the conditional mean is unique only P_X-almost everywhere and what this implies at a particular point x with P(X = x) = 0.

Explain how a regular conditional distribution or conditional density selects a pointwise version of conditioning and derive f_{Y|X}(y|x) = f_{X,Y}(x,y)/f_X(x) in the density case.

Synthesize the regression-based argument from discrete group means to continuous conditioning, Radon-Nikodym construction, and the interpretation of 'given X = x'.