Image Measures, Measures with Densities, and Change of Variables

Let $(X,\mathcal{F},\mu)$ be a measure space and let $(S,\mathcal{S})$ be a measurable space. Measurability of maps between measurable spaces is that of Measurable Function and Real-Valued Measurable Function; measurability and integrals of $[0,\infty]$-valued functions are those of Lebesgue Integral of a Nonnegative Measurable Function, with the $[0,\infty]$ conventions of Measure, Measure Space, and Probability Measure extended by the multiplication conventions $0\cdot\infty=\infty\cdot0=0$ and $a\cdot\infty=\infty\cdot a=\infty$ for $0<a\le\infty$; and integrable means integrable.

1. (Image measure) Let $T:X\to S$ be measurable with respect to $\mathcal{F}$ and $\mathcal{S}$. Then

defines a measure $\mu_T$ on $(S,\mathcal{S})$, called the image measure of $\mu$ under $T$, and $\mu_T(S)=\mu(X)$; in particular $\mu_T$ is a probability measure whenever $\mu$ is.

2. (Change of variables) In the setting of claim 1, for every measurable $g:S\to[0,\infty]$,

and a measurable $g:S\to\mathbb{R}$ is integrable with respect to $\mu_T$ if and only if $g\circ T$ is integrable with respect to $\mu$, in which case the displayed identity holds in $\mathbb{R}$.

3. (Measure with a density) Let $h:X\to[0,\infty)$ be measurable. Then

with the indicator function $\mathbf{1}_{A}$, defines a measure $\nu_h$ on $(X,\mathcal{F})$, called the measure with density $h$ with respect to $\mu$. For every measurable $f:X\to[0,\infty]$,

the pointwise product $fh$ being understood with the multiplication conventions above; and a measurable $f:X\to\mathbb{R}$ is integrable with respect to $\nu_h$ if and only if $fh$ is integrable with respect to $\mu$, in which case