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

Measures — Sheldon Axler

Mathematics — Measure Theory · measurable-function-condition-open-rays-left

Problem

Suppose (X,S)(X, \mathcal{S}) is a measurable space and f:X→Rf : X \to \mathbf{R} is a function such that f−1((−∞,a))∈Sf^{-1}((-\infty, a)) \in \mathcal{S} for all a∈Ra \in \mathbf{R}. Show that ff is an S\mathcal{S}-measurable function.

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