[[Measure theory MOC]]
# Measurable function

A measurable functions is a structure-preserving map of measurable spaces.
Let $(X,\Sigma)$ and $(Y, \mathrm{T})$ be [[Measure space|measurable spaces]].
A function $f : X \to Y$ is called **measurable** 
iff the preïmage of every measurable set is measurable[^analogous] , #m/def/measure 
i.e. $f^{-1}(E) \in \Sigma$ for any $E \in \mathrm{T}$.

[^analogous]: Note this is analogous to the topological definition of [[Continuity]].

## Properties

1. A measurable function from a measure space induces a [[Pushforward measure]] on its codomain.

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