Measure, Integration & Real Analysis · Problem 2B.27

Measures — Sheldon Axler

Mathematics — Measure Theory · measurable-functions

Problem

Prove or give a counterexample: If (X,S)(X, \mathcal{S}) is a measurable space and f:X→[−∞,∞]f : X \to [-\infty, \infty] is a function such that f−1((a,∞))∈Sf^{-1}\big((a, \infty)\big) \in \mathcal{S} for every a∈Ra \in \mathbf{R}, then ff is an S\mathcal{S}-measurable function.
Intuitive explanation

The key insight is that measurability requires the preimage of all Borel sets to be measurable, not just open intervals. You'll need to check whether the given condition—preimages of (a,∞)(a, \infty) for all real aa—is sufficient to generate all the Borel sets you need. Think about what other sets you can build from these half-open intervals using countable unions and complements, and whether there's a Borel set whose preimage might escape S\mathcal{S}.

Intuitive explanation

The key insight is that measurability requires the preimage of all Borel sets to be measurable, not just open intervals. You'll need to check whether the given condition—preimages of (a,∞)(a, \infty) for all real aa—is sufficient to generate all the Borel sets you need. Think about what other sets you can build from these half-open intervals using countable unions and complements, and whether there's a Borel set whose preimage might escape S\mathcal{S}.

Related theory

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