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integral_over_a_subset [2013/09/06 22:45] nikolaj |
integral_over_a_subset [2014/02/25 20:51] nikolaj |
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| ===== Integral over a subset ===== | ===== Integral over a subset ===== | ||
| - | ==== Definition ==== | + | ==== Set ==== |
| | @#88DDEE: $\mathbb K = \overline{\mathbb R}\lor \mathbb C$ | | | @#88DDEE: $\mathbb K = \overline{\mathbb R}\lor \mathbb C$ | | ||
| | @#88DDEE: $\langle X,\Sigma,\mu_X\rangle$ ... measure space | | | @#88DDEE: $\langle X,\Sigma,\mu_X\rangle$ ... measure space | | ||
| - | | @#88DDEE: $S\subset X$ | | ||
| - | | @#FFBB00: $\int_S: (X\to \mathbb K)\to \mathbb K$ | | + | | @#FFBB00: $\int_S: \mathcal P(X)\to(X\to \mathbb K)\to \mathbb K$ | |
| - | | $f: X\to \mathbb K$ | | + | | @#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: $\int_S\ f\ \mathrm d\mu_X:=\int_X\ f\cdot \chi_S\ \mathrm d\mu_X$ | | ||
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| ==== Parents ==== | ==== Parents ==== | ||
| - | === Requirements === | + | === Context === |
| [[Pointwise function product]], [[Characteristic function]] | [[Pointwise function product]], [[Characteristic function]] | ||
| === Refinement of === | === Refinement of === | ||
| [[Function integral]] | [[Function integral]] | ||
