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

Product Measures — Sheldon Axler

Mathematics — Measure Theory · uniqueness-markov-equality

Problem

Suppose (X,S,μ)(X, \mathcal{S}, \mu) is a measure space. Suppose h∈L1(μ)h \in \mathcal{L}^1(\mu) and ∥h∥1>0\|h\|_1 > 0. Prove that there is at most one number c∈(0,∞)c \in (0, \infty) such that μ({x∈X:∣h(x)∣≥c})=1c∥h∥1.\mu(\{x \in X : |h(x)| \geq c\}) = \frac{1}{c}\|h\|_1.

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