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 $[c,d]$
$\int_d^c\ f\ \mathrm d\mu_X\equiv -\int_c^d\ f\ \mathrm d\mu_X$ |