What is the Haar measure on the unit sphere?

Question

I try to understand the proof of Lemma 4.2. in the paper 'The Euler equations as a differential inclusion' by De Lellis and Székelyhidi. In the proof they use the Haar measure on the unit sphere $\mathbb{S}^{n-1}$ in $\mathbb{R}^n$. How is the Haar measure defined here?

Answer 1

See Section 2.1 in these lecture notes and especially Remark 2.6. One can construct the Haar measure as the normalized area of a subset of $\mathbb{S}^{n-1}$.

To randomly choose a point from a distribution that is uniform on $\mathbb{S}^{n-1}$ according to the Haar measure, it suffices to take $n$ normally-distributed random variables $(X_1,\cdots,X_n)$ and choose the point to correspond to the normalized vector $$\frac{(X_1,\cdots,X_n)}{\sqrt{\sum_{i=1}^n X_i^2}}$$