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integral_over_a_subset [2014/02/25 20:51] nikolaj |
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| - | ===== Integral over a subset ===== | ||
| - | ==== Set ==== | ||
| - | | @#88DDEE: $\mathbb K = \overline{\mathbb R}\lor \mathbb C$ | | ||
| - | | @#88DDEE: $\langle X,\Sigma,\mu_X\rangle$ ... measure space | | ||
| - | | @#FFBB00: $\int_S: \mathcal P(X)\to(X\to \mathbb K)\to \mathbb K$ | | ||
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| - | | @#DDDDDD: $f: X\to \mathbb K$ | | ||
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| - | | @#FFBB00: $\int_S\ f\ \mathrm d\mu_X:=\int_X\ f\cdot \chi_S\ \mathrm d\mu_X$ | | ||
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| - | ==== Discussion ==== | ||
| - | If $X=\mathbb R$, $a,b\in \mathbb R$, $a<b$ and the measure $\mu_X$ is such that single points have zero measure $\mu_X(\{a\})=\mu_X(\{b\})=0$ (like the standard [[Lebesgue measure]]), then we write | ||
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| - | ^ $\int_a^b\ f\ \mathrm d\mu_X\equiv\int_{[a,b]}\ f\ \mathrm d\mu_X$ ^ | ||
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| - | The zero measure of $a,b$ guaranties that we replace integrals over $[a,b)$, $(a,b]$ and $(a,b)$ by this one. | ||
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| - | If $c,d\in \mathbb R$ are numbers with $c<d$, then if we write integral symbol $\int_d^c$ (notice the switched positions of $c$ and $d$ w.r.t. their ordering) we mean the negative of the integral over $[c,d]$ | ||
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| - | ^ $\int_d^c\ f\ \mathrm d\mu_X\equiv -\int_c^d\ f\ \mathrm d\mu_X$ ^ | ||
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| - | ==== Parents ==== | ||
| - | === Context === | ||
| - | [[Pointwise function product]], [[Characteristic function]] | ||
| - | === Refinement of === | ||
| - | [[Function integral]] | ||
