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function_integral [2013/09/20 09:26] nikolaj |
function_integral [2015/06/20 16:17] nikolaj |
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| ===== Function integral ===== | ===== Function integral ===== | ||
| - | ==== Definition ==== | + | ==== Set ==== |
| - | | @#88DDEE: $\mathbb K = \overline{\mathbb R}\lor \mathbb C$ | | + | | @#55CCEE: context | @#55CCEE: $\mathbb K = \overline{\mathbb R}\lor \mathbb C$ | |
| - | | @#88DDEE: $\langle X,\Sigma,\mu\rangle\in \mathrm{MeasureSpace}(X)$ | | + | | @#55CCEE: context | @#55CCEE: $\langle X,\Sigma,\mu\rangle\in \mathrm{MeasureSpace}(X)$ | |
| - | + | | @#FFBB00: definiendum | @#FFBB00: $\int_X: (X\to \mathbb K)\to \mathbb K$ | | |
| - | | @#FFBB00: $\int_X: (X\to \mathbb K)\to \mathbb K$ | | + | | @#FFBB00: definiendum | @#FFBB00: $\int_X\ f\ \mathrm d\mu:=\int_X\ (\mathrm{Re}f)^+\ \mathrm d\mu-\int_X\ (\mathrm{Re}f)^-\ \mathrm d\mu+i\ \left( \int_X\ (\mathrm{Im}f)^+\ \mathrm d\mu-\ \int_X\ (\mathrm{Im}f)^-\ \mathrm d\mu \right)$ | |
| - | | @#FFBB00: $\int_X\ f\ \mathrm d\mu:=\int_X\ (\mathrm{Re}f)^+\ \mathrm d\mu-\int_X\ (\mathrm{Re}f)^-\ \mathrm d\mu+i\ \left( \int_X\ (\mathrm{Im}f)^+\ \mathrm d\mu-\ \int_X\ (\mathrm{Im}f)^-\ \mathrm d\mu \right)$ | | + | |
| Notice that the integral on the right hand side here is that for positive measurable numerical functions. | Notice that the integral on the right hand side here is that for positive measurable numerical functions. | ||
| - | ==== Discussion ==== | + | |
| - | ==== Parents ==== | + | ----- |
| - | === Requirements === | + | === Context === |
| [[Positive function integral]], | [[Positive function integral]], | ||
| [[Positive part of a function]], | [[Positive part of a function]], | ||
