Measure, Integration & Real Analysis · Problem 4B.1

Product Measures — Sheldon Axler

Mathematics — Measure Theory · lebesgue-differentiation-average-deviation

Problem

Suppose

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

. Prove that

\lim_{t \downarrow 0} \frac{1}{2t} \int_{b-t}^{b+t} |f - f_{[b-t,\, b+t]}| = 0

for almost every

b \in \mathbf{R}

. [Here

f_I = \frac{1}{|I|}\int_I f

denotes the average of

f

on the interval

I