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 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.

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

Measure theoretic pigeonhole principle

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Measure theoretic pigeonhole principle

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