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positive_function_integral [2013/08/18 20:23] nikolaj |
positive_function_integral [2014/03/21 11:11] (current) |
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| ===== Positive function integral ===== | ===== Positive function integral ===== | ||
| - | ==== Definition ==== | + | ==== Set ==== |
| - | | @#88DDEE: $M \in \mathrm{MeasureSpace}(X)$ | | + | | @#55CCEE: context | @#55CCEE: $M \in \mathrm{MeasureSpace}(X)$ | |
| - | | @#55EE55: $\int_X:\mathcal M^+\to \mathbb R_+$ | | + | | @#55EE55: postulate | @#55EE55: $\int_X:\mathcal M^+\to \mathbb R_+$ | |
| | $ f\uparrow u_n$ | $u_n\in \mathcal T^+$ | | | $ f\uparrow u_n$ | $u_n\in \mathcal T^+$ | | ||
| - | | @#55EE55: $\int_X\ f\ \mathrm d\mu:=\mathrm{lim}_{n\to \infty}\int_X\ u_n\ \mathrm d\mu$ | | + | | @#55EE55: postulate | @#55EE55: $\int_X\ f\ \mathrm d\mu:=\mathrm{lim}_{n\to \infty}\int_X\ u_n\ \mathrm d\mu$ | |
| 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**: |
| - | === Requirements === | + | |
| - | [[Step function integral]], | + | If $f_n$ is a [[growing sequence]] in $\mathcal M^+$, we have |
| - | [[Positive measurable numerical function]], | + | |
| - | [[Growing sequence]] | + | ^ $\int_X\left(\mathrm{lim}_{n\to\infty}f_n\right)\mathrm d\mu=\mathrm{lim}_{n\to\infty}\int_X f_n\mathrm d\mu$ ^ |
| + | ==== Parents ==== | ||
| + | === Context === | ||
| + | [[Growing sequence]], [[Step function integral]], | ||
| + | [[Positive measurable numerical function]] | ||
