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positive_function_integral [2013/09/06 23:32]
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. ​
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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]], [[Step function integral]], ​ [[Growing sequence]], [[Step function integral]], ​
 [[Positive measurable numerical function]] [[Positive measurable numerical function]]
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