Measure, Integration & Real Analysis · Problem 4B.3

Product Measures — Sheldon Axler

Mathematics — Measure Theory · lebesgue-differentiation-l2

Problem

Suppose

f: \mathbf{R} \to \mathbf{R}

is a Lebesgue measurable function such that

f^2 \in \mathcal{L}^1(\mathbf{R})

. Prove that

\lim_{t \downarrow 0} \frac{1}{2t} \int_{b-t}^{b+t} |f - f(b)|^2 = 0

for almost every

b \in \mathbf{R}