Measurable functions
Objective. Use preimages and approximation by simple functions.
- 1. Learn
- 2. Worked example
- 3. Practice
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Learn: the key idea
A function f:X→ℝ is measurable with respect to a σ-algebra 𝒜 if the preimage f⁻¹((a,∞)) lies in 𝒜 for every real a, which is equivalent to preimages of all Borel sets being measurable. This preimage criterion lets us build new measurable functions from old ones: sums, products, and pointwise limits of measurable functions stay measurable. Every measurable function is a pointwise limit of simple functions (finite linear combinations of indicator functions), which is the key approximation tool for defining integrals. For instance, f(x)=x² is measurable because f⁻¹((a,∞)) is an open set, hence Borel, for every a.
Worked example
Decide whether f(x)=⌊x⌋ (floor function) is measurable with respect to the Borel σ-algebra on ℝ.
- 1Need f⁻¹((a,∞)) Borel for all a
- 2For a=2, f⁻¹((2,∞)) = [3,∞), a half-line
- 3Half-lines are Borel sets, and this holds for every a similarly
- 4So f is measurable
Answer: Yes, f is measurable
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Start practisingCommon mistake. Students assume continuity is needed for measurability, but measurable functions can be highly discontinuous; indicator functions of Borel sets are measurable yet typically discontinuous everywhere on the boundary. Check the preimage condition, not continuity.
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