Measure theoretic pigeonhole principle

Let $𝑋$ and $𝑌$ be measure spaces and $𝑓 : 𝑋 \to 𝑌$ be a measurable function. We call $𝑓$ finitely piecewise measure-preserving iff there exists a partition ${𝑋 𝑖} 𝑛 𝑖 = 1$ into measurable sets such that

𝜇 (𝑓 (𝑋 𝑖)) = 𝜇 (𝑋 𝑖)

for all $𝑖 \in ℕ 𝑛$ . Given such a function, if $𝜇 (𝑋) > 𝜇 (𝑓 (𝑋))$ , then $𝑓$ is not injective.¹ #m/thm/measure

Proof

If $𝑓$ is injective then $𝑓 (𝑋)$ is the disjoint union of the $𝑓 (𝑋 𝑖)$ and we have

𝜇 (𝑓 (𝑋)) = 𝑛 \sum 𝑖 = 1 𝜇 (𝑓 (𝑋 𝑖)) = 𝑛 \sum 𝑖 = 1 𝜇 (𝑋 𝑖) = 𝜇 (𝑋)

so the above condition suffices for non-injectivity.

#state/tidy | #lang/en | #SemBr

2022. Algebraic number theory course notes, ¶3.4, p. 61 ↩

Measure theoretic pigeonhole principle

Footnotes