Measure Theory // 2

Define absolute continuity of one measure with respect to another (μ << ν) and explain its role as a prerequisite for defining a density.

State the Radon-Nikodym theorem, identifying the conditions for its application and the resulting Radon-Nikodym derivative dμ/dν.

Interpret the Radon-Nikodym derivative as a generalized density of one measure with respect to another.

Relate the Radon-Nikodym derivative to a standard probability density function by showing the PDF is the derivative with respect to Lebesgue measure.

Review the definition of conditional expectation with respect to a σ-algebra, E[Y|σ(X)], as the formal solution to the conditioning paradox.

Explain how the Radon-Nikodym theorem provides the concrete mechanism for constructing E[Y|σ(X)].

For a bivariate normal model, derive the conditional distribution of an asset return given an index value using the elementary formula f(y|x) = f(x,y)/f_X(x).

Apply the measure-theoretic framework to the bivariate normal model by identifying the appropriate measures (joint, marginal) and verifying absolute continuity.

Compute the Radon-Nikodym derivative for the bivariate normal case to find the conditional expectation and verify that it reproduces the result from the elementary formula.

Define a regular conditional distribution as the formal object that rigorously represents 'the distribution of Y given X=x'.

Synthesize the complete logical argument for the essay: from the choice of continuous models to the resolution of the conditioning paradox via measure theory.