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positive_function_integral [2013/09/06 23:32] nikolaj |
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| - | ===== Positive function integral ===== | ||
| - | ==== Definition ==== | ||
| - | | @#88DDEE: $M \in \mathrm{MeasureSpace}(X)$ | | ||
| - | | @#55EE55: $\int_X:\mathcal M^+\to \mathbb R_+$ | | ||
| - | |||
| - | | $ f\uparrow u_n$ | $u_n\in \mathcal T^+$ | | ||
| - | |||
| - | | @#55EE55: $\int_X\ f\ \mathrm d\mu:=\mathrm{lim}_{n\to \infty}\int_X\ u_n\ \mathrm d\mu$ | | ||
| - | |||
| - | Notice that the integral on the right hand side here is that for positive real step functions. | ||
| - | ==== Discussion ==== | ||
| - | **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]], [[Step function integral]], | ||
| - | [[Positive measurable numerical function]] | ||
