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integral_over_a_subset [2013/09/06 22:45] nikolaj |
integral_over_a_subset [2014/03/21 11:11] (current) |
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| ===== Integral over a subset ===== | ===== Integral over a subset ===== | ||
| - | ==== Definition ==== | + | ==== Set ==== |
| - | | @#88DDEE: $\mathbb K = \overline{\mathbb R}\lor \mathbb C$ | | + | | @#55CCEE: context | @#55CCEE: $\mathbb K = \overline{\mathbb R}\lor \mathbb C$ | |
| - | | @#88DDEE: $\langle X,\Sigma,\mu_X\rangle$ ... measure space | | + | | @#55CCEE: context | @#55CCEE: $\langle X,\Sigma,\mu_X\rangle$ ... measure space | |
| - | | @#88DDEE: $S\subset X$ | | + | |
| - | | @#FFBB00: $\int_S: (X\to \mathbb K)\to \mathbb K$ | | + | | @#FFBB00: definiendum | @#FFBB00: $\int_S: \mathcal P(X)\to(X\to \mathbb K)\to \mathbb K$ | |
| - | | $f: X\to \mathbb K$ | | + | | @#DDDDDD: range | @#DDDDDD: $f: X\to \mathbb K$ | |
| - | | @#FFBB00: $\int_S\ f\ \mathrm d\mu_X:=\int_X\ f\cdot \chi_S\ \mathrm d\mu_X$ | | + | | @#FFBB00: definiendum | @#FFBB00: $\int_S\ f\ \mathrm d\mu_X:=\int_X\ f\cdot \chi_S\ \mathrm d\mu_X$ | |
| ==== Discussion ==== | ==== Discussion ==== | ||
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| ==== Parents ==== | ==== Parents ==== | ||
| === Requirements === | === Requirements === | ||
| - | [[Pointwise function product]], [[Characteristic function]] | + | [[Pointwise function product]], [[Characteristic function]], [[Function integral]] |
| - | === Refinement of === | + | |
| - | [[Function integral]] | + | |
