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positive_function_integral [2013/09/06 22:04] 127.0.0.1 external edit |
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. | ||
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^ $\int_X\left(\mathrm{lim}_{n\to\infty}f_n\right)\mathrm d\mu=\mathrm{lim}_{n\to\infty}\int_X f_n\mathrm d\mu$ ^ | ^ $\int_X\left(\mathrm{lim}_{n\to\infty}f_n\right)\mathrm d\mu=\mathrm{lim}_{n\to\infty}\int_X f_n\mathrm d\mu$ ^ | ||
==== Parents ==== | ==== Parents ==== | ||
- | === Requirements* === | + | === Context === |
- | [[Growing sequence]] | + | [[Growing sequence]], [[Step function integral]], |
- | === Requirements === | + | |
- | [[Step function integral]], | + | |
[[Positive measurable numerical function]] | [[Positive measurable numerical function]] |