Applying Fubini's theorem for spherical coordinates

I want to prove that if $B^3_R$ is a 3-ball of radius $R$ around $0$, then

$$ \int_{B^3_R}||x||^p dx = 4\pi\frac{R^{p+3}}{p+3}$$

I of want to use Fubini's theorem here to switch to spherical coordinates, but the part that is confusing me is the following: Fubini's theorem talks about the product measure of two spaces, but if I understand correctly the Lebesgue measure on a product is not quite the product measure. How relevant is this, and what does a formal proof look like?

Thanks a lot!

Solution 1:

In general suppose you have a set $X$ with two $\sigma$-algebras, $\mathfrak{M}\subset \mathfrak{N}$ and a measure $\nu$ on the larger one, and $\mu$ is the resitriction to the smaller one. Suppose $f:X\to\Bbb{C}$ is $\mathfrak{M}$-$\mathcal{B}(\Bbb{C})$ measurable (i.e with respect to the smaller $\sigma$-algebra), then it is almost immediate from definitions that for any set $E\in\mathfrak{M}$ in the smaller $\sigma$-algebra, $f$ is integrable over $E$ with respect to $\mu$ if and only if it is with respect to $\nu$, in which case \begin{align} \int_Ef\,d\mu&=\int_Ef\,d\nu. \end{align} Unwind the definitions: clearly true for indicators of sets in $\mathfrak{M}$, hence for non-negative linear combinations, hence all non-negative $\mathfrak{M}$-measurable functions, hence all integrable ones.

For each integer $n\geq 1$, let $\mathcal{L}_n,\mathcal{B}_n, \lambda_n,\beta_n$ be the Lebesgue, Borel $\sigma$-algebras and measures respectively, i.e $\beta_n:=\lambda_n|_{\mathcal{B}_n}$. Then, we have $\beta_n= (\beta_1)^{\otimes n}$ is the $n$-fold product measure (and $\mathcal{B}_n=(\mathcal{B}_1)^{\otimes n}$ because $\Bbb{R}$ is second-countable. See Folland's Chapter 1 for these details).

Because of these facts, after changing variables, you can use Fubini naively on the Borel sets. Anyway, you should note that Folland also states (and outlines as an exercise) a version of Fubini valid for complete measures (i.e $\mu,\nu$ $\sigma$-finite and complete, and $\lambda$ the completion of $\mu\otimes \nu$), so take a look at his book for that. Also, the general integral calculation becomes almost obvious now: for $n$-dimensional balls of radius $R$, and $p>-n$, \begin{align} \int_{B_R^n(0)}\|x\|^p_2\,dx&=A_{n-1}\frac{R^{p+n}}{p+n}=\frac{2\pi^{n/2}}{\Gamma(n/2)}\frac{R^{p+n}}{p+n}. \end{align}

One thing to note is that, as mentioned in the comments, Folland introduces the spherical measure in a rather ad-hoc/reversed way. The way I prefer to think of the surface measure is as the positive measure induced by the Riemannian metric on $S^{n-1}$ induced from $\Bbb{R}^{n}$. Also, Folland's text has a statement and proof of the change of variables theorem for Lebesgue integrals, so you can look at that as well (if you have access to extra measure-theoretic mumbo jumbo, i.e Chapter 3 of Folland/chapter 6 of Rudin's RCA, you can also follow along this answer I wrote)

Solution 2:

Spherical coordinates in $\mathbb R^3$: $$ dx_1\,dx_2\,dx_3=r^2\,dr\,d\omega $$ where $\int_{\|x\|=1}d\omega=4\pi$, the area of a unit sphere.

Hence $$ \int_{\|x\|\le R}\|x\|^p\,dx_1\,dx_2\,dx_3=\int_0^R\left(\int_{\|x\|=1}r^pr^2\,d\omega\right)\,dr=\int_0^R 4\pi r^{p+2}dr=\frac{4\pi R^{p+3}}{p+3} $$