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 positive_function_integral [2013/09/06 23:32]nikolaj positive_function_integral [2014/03/21 11:11] (current) Both sides previous revision Previous revision 2013/09/06 23:32 nikolaj 2013/09/06 22:04 external edit2013/08/19 16:45 nikolaj 2013/08/18 20:25 nikolaj 2013/08/18 20:23 nikolaj 2013/08/18 20:20 nikolaj 2013/08/18 20:15 nikolaj 2013/08/18 20:14 nikolaj old revision restored (2013/08/18 19:53)2013/08/18 20:10 nikolaj 2013/08/18 19:57 nikolaj 2013/08/18 19:53 nikolaj created 2013/09/06 23:32 nikolaj 2013/09/06 22:04 external edit2013/08/19 16:45 nikolaj 2013/08/18 20:25 nikolaj 2013/08/18 20:23 nikolaj 2013/08/18 20:20 nikolaj 2013/08/18 20:15 nikolaj 2013/08/18 20:14 nikolaj old revision restored (2013/08/18 19:53)2013/08/18 20:10 nikolaj 2013/08/18 19:57 nikolaj 2013/08/18 19:53 nikolaj created Line 1: Line 1: ===== 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. ​ Line 17: Line 17: ^ $\int_X\left(\mathrm{lim}_{n\to\infty}f_n\right)\mathrm d\mu=\mathrm{lim}_{n\to\infty}\int_X f_n\mathrm d\mu$ ^ ^ $\int_X\left(\mathrm{lim}_{n\to\infty}f_n\right)\mathrm d\mu=\mathrm{lim}_{n\to\infty}\int_X f_n\mathrm d\mu$ ^ ==== Parents ==== ==== Parents ==== - === Requirements ​=== + === Context ​=== [[Growing sequence]], [[Step function integral]], ​ [[Growing sequence]], [[Step function integral]], ​ [[Positive measurable numerical function]] [[Positive measurable numerical function]]