Scheffe's Lemma

Last updated Nov 1, 2022

Let $F_{1}, F_{2}$ be two Probability Distribution Functions and suppose they have density functions $f_{1}, f_{2}$ with respect to Measure $\mu$ on $\mathcal{B}( \mathbb{R})$. Then

$$\sup_{B \in \mathcal{B}(\mathbb{R})} |F_{1}(B) - F_{2}(B)| = \frac{1}{2} \int |f_{1} - f_{2}| d \mu$$ In particular, if $F_{n}$ is a sequence of Probability Distribution Functions, $f_{n}$ the corresponding Probability Density Functions, and we have target $F$ with density $f$, then if $f_{n} \to f$ $\mu$-a.e. we have that $F_{n} \to F$ in Total Variation.

TODO

• Borel Sigma Algebra
• Real Numbers

• 2022-02-25 - Resnick - A Probability Path - pg 253