pages/measure-and-integration/measure-theory/measurable-maps.md


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---
title: Measurable Maps
parent: Measure Theory
grand_parent: Measure and Integration
nav_order: 3
---

# {{ page.title }}

{% definition Measurable Map %}
Suppose $(X,\mathcal{A})$ and $(Y,\mathcal{B})$ are measurable spaces.
We say that a map $f: X \to Y$ is *measurable* (with respect to $\mathcal{A}$ and $\mathcal{B}$) if
$f^{-1}(B) \in \mathcal{A}$ for all $B \in \mathcal{B}$.
{% enddefinition %}

{% proposition %}
The composition of measurable maps is measurable.
{% endproposition %}

It is sufficient to check measurability for a generator:

{% proposition %}
Suppose that $(X,\mathcal{A})$ and $(Y,\mathcal{B})$ are measurable spaces,
and that $\mathcal{E}$ is a generator of $\mathcal{B}$.
Then a map $f : X \to Y$ is measurable iff
$f^{-1}(E) \in \mathcal{A}$ for every $E \in \mathcal{E}$.
{% endproposition %}