Pigeonhole principle

Measure theoretic pigeonhole principle

Let and be measure spaces and be a measurable function. We call finitely piecewise measure-preserving iff there exists a partition into measurable sets such that

for all . Given such a function, if , then is not injective.¹ measure

Proof

If is injective then is the disjoint union of the and we have

so the above condition suffices for non-injectivity.

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tidy | en | sembr

Footnotes

1. 2022. Algebraic number theory course notes, ¶3.4, p. 61 ↩