I asked a question here last day. The answer seems to be too esoteric for me to understand. Please explain so that a physics student like me can understand.
Let $(X,\mu)$ be a measures set, let $f:X\rightarrow \mathbb R$ be a measurable function and let $g:Y\rightarrow X$ be change of variables, then \begin{align} \int_{X} |f\circ g|\, \mathrm d \mu< +\infty \quad\Leftrightarrow\quad \int_{Y} |f|\,g\#\mathrm d \mu < +\infty \tag1 \end{align}
This theorem is valid in a very wide generality : the change of variable $g$ do not need to be continuous or bijective and it includes all change of variable you might encounter. I wrote $\phi \# \mathrm d \mu$ for the image measure. For instance, let $X=\mathbb R^3$ and $Y= [0,2\pi]\times [0,\pi]\times \mathbb R_ +^*$, let $$ g : \begin{matrix}Y&\rightarrow &X \\ (\theta,\phi,r)&\mapsto& (r\cos(\theta)\cos(\phi),r\sin(\theta)\cos(\phi),r\sin(\phi))\end{matrix}$$ You recognize your spherical change of variable and $$\mathrm d\mu = \mathrm d x\mathrm dy \mathrm d z \quad\quad \quad g\#\mathrm d \mu = r^2\sin(\phi)\mathrm d \theta \mathrm d \phi \mathrm d r$$ here $f=\rho \hat r$.
Please write equation (1) in the language of elementary multivariable calculus