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

Product Measures — Sheldon Axler

Mathematics — Measure Theory · markov-inequality-generalization

Problem

Suppose

(X, \mathcal{S}, \mu)

is a measure space and

h: X \to \mathbf{R}

is an

\mathcal{S}

-measurable function. Prove that

\mu(\{x \in X : |h(x)| \geq c\}) \leq \frac{1}{c^p} \int |h|^p \, d\mu

for all positive numbers

c

and

p