From 28407333ffceca9b99fae721c30e8ae146a863da Mon Sep 17 00:00:00 2001 From: Justin Gassner Date: Wed, 14 Feb 2024 07:24:38 +0100 Subject: Update --- .../measure-theory/measurable-maps.md | 27 ++++++++++++++++++++++ 1 file changed, 27 insertions(+) create mode 100644 pages/measure-and-integration/measure-theory/measurable-maps.md (limited to 'pages/measure-and-integration/measure-theory/measurable-maps.md') diff --git a/pages/measure-and-integration/measure-theory/measurable-maps.md b/pages/measure-and-integration/measure-theory/measurable-maps.md new file mode 100644 index 0000000..5b7a76e --- /dev/null +++ b/pages/measure-and-integration/measure-theory/measurable-maps.md @@ -0,0 +1,27 @@ +--- +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 %} -- cgit v1.2.3-54-g00ecf