Measure, Integration & Real Analysis · Problem 4B.4

Product Measures — Sheldon Axler

Mathematics — Measure Theory · lebesgue-differentiation-weaker-hypothesis

Problem

Prove that the Lebesgue Differentiation Theorem (4.19) still holds if the hypothesis that

\int_{-\infty}^{\infty} |f| < \infty

is weakened to the requirement that

\int_{-\infty}^{x} |f| < \infty

for all

x \in \mathbf{R}