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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?

Ben Steffan
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Lae
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    My guess would be treating it as a homogeneous space of the special orthogonal group: https://en.wikipedia.org/wiki/Haar_measure#Measures_on_homogeneous_spaces – Matt Samuel Jun 15 '20 at 15:24
  • Thank you for the hint! If I understood it correctly, this means that $\int_{\mathbb{S}^{n-1}} f(Av) : dv = \int_{\mathbb{S}^{n-1}} f(v) : dv$ holds for all $A\in SO(n)$ and all integrable functions on $\mathbb{S}^{n-1}$. In the paper they then conclude $ \int_{\mathbb{S}^{n-1}}(v,v\otimes v - \frac{I_{n}}{n}) : dv = 0$. Does this mean that there exists an $A\in SO(n)$ such that $(Av,Av\otimes Av - \frac{I_{n}}{n}) = 0$? – Lae Jun 16 '20 at 11:01

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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 $$\frac{(X_1,\cdots,X_n)}{\sqrt{\sum_{i=1}^n X_i^2}}$$

Quantum Mechanic
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