Let $\boldsymbol{r}$ represent a point of $\mathbb{R}^3$ with two components fixed and one, say $r_k$, free to vary on $[a,b]$, and let $V\subset\mathbb{R}^3$ be a measurable, according to the tridimensional Lebesgue measure, and bounded subset.
I am trying to understand whether the Lebesgue integral $$\int_V\int_{[a,b]}\frac{1}{\|\boldsymbol{r}-\boldsymbol{l}\|^2}d\mu_{r_k}d\mu_{\boldsymbol{l}}$$ $$=\int_V\int_{[a,b]}\frac{1}{\sqrt{(r_i-l_i)^2+(r_j-l_j)^2+(r_k-l_k)^2}}d\mu_{r_k}d\mu_{\boldsymbol{l}}$$or equivalently$$\int_{[a,b]}\int_V\frac{1}{\|\boldsymbol{r}-\boldsymbol{l}\|^2}d\mu_{\boldsymbol{l}}d\mu_{r_k}\quad\text{ or }\quad\int_{[a,b]\times V}\frac{1}{\|\boldsymbol{r}-\boldsymbol{l}\|^2}d\mu_{\boldsymbol{l}}\otimes\mu_{r_k}$$exist finite. All the measures are intended as the usual $n$-dimensional Lebesgue measures. Does it exist and, if it does, how can it be proved?
I do see that, for all fixed $\boldsymbol{r}$, $\int_V\frac{1}{\|\boldsymbol{r}-\boldsymbol{l}\|^2}d\mu_{\boldsymbol{l}}$ exists finite by chosing a ball $B(\boldsymbol{r},R)\supset V$ and calculating $$\int_V\frac{1}{\|\boldsymbol{r}-\boldsymbol{l}\|^2}d\mu_{\boldsymbol{l}}\le\int_{B(\boldsymbol{r},R)}\frac{1}{\|\boldsymbol{r}-\boldsymbol{l}\|^2}d\mu_{\boldsymbol{l}}=\lim_{\varepsilon\to 0}\,\mathscr{R}\int_{B(\boldsymbol{r},R)\setminus B(\boldsymbol{r},\varepsilon)}\frac{1}{\|\boldsymbol{r}-\boldsymbol{l}\|^2}d^3l=4\pi R$$ where I use the notation $\mathscr{R}$ for the Riemann integral, which I have calculated by using spherical coordinates. But here, $r_k$ is free to vary in $[a,b]$, which prevents me from using this argument. I heartily thank any answerer.