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real_step_function [2013/09/06 22:04] 127.0.0.1 external edit |
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| - | ===== Real step function ===== | ||
| - | ==== Definition ==== | ||
| - | | @#88DDEE: $\langle X,\Sigma\rangle\in\mathrm{MeasurableSpace}(X)$ | | ||
| - | | @#55EE55: $f\in \mathcal T$ | | ||
| - | |||
| - | | @#88DDEE: $f\in\mathrm{Measurable}(X,\mathbb R)$ | | ||
| - | |||
| - | Where we consider the Borel algebra over $\mathbb R$ | ||
| - | |||
| - | | @#55EE55: $\mathrm{im}(f)$...[[glossary|finite]] | | ||
| - | |||
| - | ==== Discussion ==== | ||
| - | These functions can be written as | ||
| - | |||
| - | $f=\sum_{j=1}^n\alpha_j\cdot\chi_{E_n}$ | ||
| - | |||
| - | with $\alpha_j$'s real numbers and $E_n$'s in the measurable algebra. | ||
| - | ==== Parents ==== | ||
| - | === Refinement of === | ||
| - | [[Measurable function]] | ||
| - | === Requirements === | ||
| - | [[Real number]] | ||
