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

Product Measures — Sheldon Axler

Mathematics — Measure Theory · chebyshev-inequality

Problem

Suppose (X,S,μ)(X, \mathcal{S}, \mu) is a measure space with μ(X)=1\mu(X) = 1 and h∈L1(μ)h \in \mathcal{L}^1(\mu). Prove that μ ⁣({x∈X:∣h(x)−∫h dμ∣≥c})≤1c2(∫h2 dμ−(∫h dμ)2)\mu\!\left(\left\{x \in X : \left|h(x) - \int h \, d\mu\right| \geq c\right\}\right) \leq \frac{1}{c^2}\left(\int h^2 \, d\mu - \left(\int h \, d\mu\right)^2\right) for all c>0c > 0. [The result above is called Chebyshev's inequality; it plays an important role in probability theory. Pafnuty Chebyshev (1821–1894) was Markov's thesis advisor.]

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