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author | Justin Gassner <justin.gassner@mailbox.org> | 2024-02-14 07:24:38 +0100 |
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committer | Justin Gassner <justin.gassner@mailbox.org> | 2024-02-14 07:24:38 +0100 |
commit | 28407333ffceca9b99fae721c30e8ae146a863da (patch) | |
tree | 67fa2b79d5c48b50d4e394858af79c88c1447e51 /pages/measure-and-integration/lebesgue-integral/almost-everywhere.md | |
parent | 777f9d3fd8caf56e6bc6999a4b05379307d0733f (diff) | |
download | site-28407333ffceca9b99fae721c30e8ae146a863da.tar.zst |
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diff --git a/pages/measure-and-integration/lebesgue-integral/almost-everywhere.md b/pages/measure-and-integration/lebesgue-integral/almost-everywhere.md new file mode 100644 index 0000000..a77cf9a --- /dev/null +++ b/pages/measure-and-integration/lebesgue-integral/almost-everywhere.md @@ -0,0 +1,27 @@ +--- +title: Almost Everywhere +parent: Lebesgue Integral +grand_parent: Measure and Integration +nav_order: 1 +--- + +# {{ page.title }} + +{% definition Almost Everywhere %} +We say that a property $P(x)$ depending on $x \in X$ +holds *almost everywhere* (abbreviated by *a.e.*) or for *almost all $x \in X$* if +the set of points where it does not hold has measure zero. +{% enddefinition %} + +In other words, $P(x)$ a.e. iff +$\mu(\set{x \in X : \neg P(x)}) = 0$. + +{% theorem %} +Let $f : X \to \overline{\RR}$ be a nonnegative measurable function. Then + +$$ +\int_X f \, d\mu = 0 +$$ + +holds if and only if $f$ vanishes almost everywhere. +{% endtheorem %} |