Created: August 27, 2022
Modified: August 27, 2022

measurable function

This page is from my personal notes, and has not been specifically reviewed for public consumption. It might be incomplete, wrong, outdated, or stupid. Caveat lector.

A function $f: \Omega \to S$ is measurable with respect to sigma-algebras $\mathcal{F}$ on its domain $\Omega$ and $\Sigma$ on its range $S$ if the pre-image of any event $B\in \Sigma$ is measurable in the domain:

f^{-1}(B) \in \mathcal{F},\qquad \forall B \in \Sigma.

In many cases, one or both of the sigma-algebras involved are defined implicitly, e.g., as the set of all open sets (the Borel sigma-algebra).

measurable function

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