I am trying to prove that the Lebesgue measure on $S^n$ is the unique countably additive, rotation-invariant measure of total measure 1 defined on Lebesgue-measurable sets.
I know the proof of the analogous statement for the Lebesgue measure with $\mathbb{R}^n$. However, in that case we start from rectangles, which have the nice property that a disjoint union of countably many rectangles cover the whole space and that the intersection of two rectangles is again a rectangle. I don't know how to define the analogue of a rectangle on the sphere, even if in $S^1$ and $S^2$ I can sort of picture what they should look like…
Also I have found no reference apart from one in Russian so if someone knows of something in English French Italian or German please answer with a link!
Best Answer
For $S^1$ you can copy the proof for $\mathbb{R}$: for every $n\ge1$ segments of length $2\pi/n$ will have measure $1/n$th of the full measure of the circle. Once you have that you can use approximations to show that a segment of length $a$ should have measure $a/(2\pi)$ of the full measure. Then you are done by uniqueness of the extension.
For $S^2$ you can do similar things: hemispheres have measure $1/2$ of the full measure. The `strips' between meridians with $a\le \text{longitude}<b$ will have measure $|b-a|/(2\pi)$ of the full measure. Using rotations this holds in all directions. Then you can deal with bands between latitudes and show that the measure of a rectangle between longitudes $a$ and $b$ will have measure $|b-a|/(2\pi)$ of the measure of the band. In this way you can build up a ring on which the measure behaves like Lebesgue measure and which is large enough to apply the uniqueness-of-extension to.