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positive_measurable_numerical_function [2013/08/18 20:02] nikolaj |
positive_measurable_numerical_function [2014/03/21 11:11] |
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- | ===== Positive measurable numerical function ===== | ||
- | ==== Definition ==== | ||
- | | @#88DDEE: $ \langle X,\Sigma_X\rangle\in \mathrm{MeasurableSpace}(X) $ | | ||
- | | @#55EE55: $f\in \mathcal M^+$ | | ||
- | |||
- | | @#88DDEE: $f\in \mathrm{Measurable}(X,\overline{\mathbb R})$ | | ||
- | | $x\in X$ | | ||
- | |||
- | | @#55EE55: $f(x)\ge 0$ | | ||
- | |||
- | ==== Discussion ==== | ||
- | For the definition of the integral, it's crucial to know that for every $f\in \mathcal M^+$, there is a sequence $u_n$ with elements in the step functions $\mathcal T^+$, with $u_n\uparrow f$. | ||
- | ==== Context ==== | ||
- | === Subset of === | ||
- | [[Measurable numerical function]] |