Let $X\sim Unif(B(0,\sqrt{n})$, a uniform random vector in the closed n-ball. I am trying to use the following expression for $X$ in a proof:
$$X=\sqrt{n}\frac{g}{\|g\|_2}r^{\frac{1}{n}}\,,$$ where $g\sim N(0,\mathbb{1}_n)$ and $r\sim Unif([0,1])$.
However, in order to show this expression is a valid representation of $X$, I of course need to show it is uniform. I understand that I am supposed to show the probability of any event is equal to the volume of the subset of $B(0,\sqrt{n})$ divided by the volume of $B(0,\sqrt{n})$. Is there some other way of showing this?
EDIT:
$B(a,b):=$ the closed n-ball centred at $a$, with radius $b$.