TITLE: Haar's theorem for the rotation-invariant distribution on the sphere
QUESTION [12 upvotes]: Let $\mu$ be a Borel probability measure on $\mathcal S^{n-1}$. Suppose $\mu$ is rotation-invariant, that is, $\mu(UA) = \mu(A)$ for any Borel set $A\subset\mathcal S^{n-1}$ and orthogonal matrix $U\in\mathbb R^{n\times n}$. Then $\mu$ is the
  uniform measure.

Could you recommend a reference (preferably a book) for this simple version of Haar's theorem?

REPLY [4 votes]: I've found an answer for my question: Theorem 3.4 of Pertti Mattila's Geometry of sets and measures in Euclidean spaces: Fractals and rectifiability.