Integral over a subset

Set

context	$\mathbb K = \overline{\mathbb R}\lor \mathbb C$
context	$\langle X,\Sigma,\mu_X\rangle$ … measure space

definiendum

$\int_S: \mathcal P(X)\to(X\to \mathbb K)\to \mathbb K$

range

$f: X\to \mathbb K$

definiendum

$\int_S\ f\ \mathrm d\mu_X:=\int_X\ f\cdot \chi_S\ \mathrm d\mu_X$

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

$\int_a^b\ f\ \mathrm d\mu_X\equiv\int_{[a,b]}\ f\ \mathrm d\mu_X$

The zero measure of $a,b$ guaranties that we replace integrals over $[a,b)$ , $(a,b]$ and $(a,b)$ by this one.

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 $[c,d]$

$\int_d^c\ f\ \mathrm d\mu_X\equiv -\int_c^d\ f\ \mathrm d\mu_X$

Parents

Requirements

Pointwise function product, Characteristic function, Function integral