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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 %}