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CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27
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Continuous functions on Euclidean spaces are Borel measurable

Statement

Assume the Axiom of Countable Choice. Let n,m≥1. Every continuous map f:Rn→Rm is Borel measurable in the sense of Borel measurable and Lebesgue measurable functions on Rn.

Facts & Assumptions

Given: The Axiom of Countable Choice, natural numbers n,m≥1, and a continuous function f:Rn→Rm.

[L1]

A continuous map has Borel preimages of Borel sets. (A continuous map has Borel preimages of Borel sets)

Proof

technique · direct
1.1

Let B⊆Rm be Borel. By [L1], the preimage [L1] f−1(B) is a Borel subset of Rn.

L1
2.1

By the definition of Borel measurability on Euclidean spaces, step 1.1 says [step 1.1] exactly that f is Borel measurable.

step 1.1∎

Depends on

Used by

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Dependency tree · two levels

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Sources