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positive_function_integral [2013/08/18 20:23] nikolaj |
positive_function_integral [2013/09/06 23:32] nikolaj |
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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 === | === Requirements === | ||
| - | [[Step function integral]], | + | [[Growing sequence]], [[Step function integral]], |
| - | [[Positive measurable numerical function]], | + | [[Positive measurable numerical function]] |
| - | [[Growing sequence]] | + | |
