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positive_function_integral [2013/08/18 20:23] nikolaj |
positive_function_integral [2013/09/06 22:04] 127.0.0.1 external edit |
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Notice that the integral on the right hand side here is that for positive real step functions. | Notice that the integral on the right hand side here is that for positive real step functions. | ||
==== Discussion ==== | ==== Discussion ==== | ||
- | ==== Context ==== | + | **Monotone convergence theorem**: |
+ | |||
+ | If $f_n$ is a [[growing sequence]] in $\mathcal M^+$, we have | ||
+ | |||
+ | ^ $\int_X\left(\mathrm{lim}_{n\to\infty}f_n\right)\mathrm d\mu=\mathrm{lim}_{n\to\infty}\int_X f_n\mathrm d\mu$ ^ | ||
+ | ==== Parents ==== | ||
+ | === Requirements* === | ||
+ | [[Growing sequence]] | ||
=== Requirements === | === Requirements === | ||
[[Step function integral]], | [[Step function integral]], | ||
- | [[Positive measurable numerical function]], | + | [[Positive measurable numerical function]] |
- | [[Growing sequence]] | + |