Measure, Integration & Real Analysis · Problem 4B.2

Product Measures — Sheldon Axler

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

Problem

Suppose

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

. Prove that

\lim_{t \downarrow 0} \sup\!\left\{\frac{1}{|I|}\int_I |f - f_I| : I \text{ is an interval of length } t \text{ containing } b\right\} = 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