The following is a simple and beautiful proof, shown to me to my great delight while I was in high school, that the -ball of radius has -volume
Although I expect most readers will know it, I believe that everybody should see it. I don’t know the history of it, and would be interested in learning.
Suppose that the -ball of radius has -volume . Then, by considering linear transformations, the -ball of radius has -volume . Moreover, differentiating this with respect to should produce the surface -volume of the -sphere : thus we expect .
Now consider the integral
We will compute in two different ways. On the one hand,
On the other hand, the form of suggests introducing a radial coordinate . Computing this way,
Suppose that is a locally compact group and that is a closed subgroup with an -left-invariant regular Borel measure such that possesses a -left-invariant regular Borel measure . For instance, , , and . The following is how you then induce a Haar measure on . (Technically, it’s easier to construct Haar measure on compact groups, so this extends that construction slightly.)
For , define by
Let . Then , as a function of is supported on , so the above integral is finite. Moreover, if and is compact, then
Because a continuous function on a compact set is uniformly continuous (in the sense that there exists a neighbourhood of such that implies ), is continous. Since is -right-invariant, descends to a continuous function defined on . Moreover, if is the quotient map , then is supported on , so . Finally, define by
This linear functional is positive (in the sense that implies ), so the Riesz representation theorem guarantees the existence of a regular Borel measure on such that
for all . It is now a simple matter to check that is -left-invariant.