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Consider the space $\mathrm{SU}(2)^\natural$ of conjugacy classes in $\mathrm{SU}(2)$. It has a natural identification with the interval $[0,\pi]$ with Haar measure $\frac{2}{\pi} \sin^2\theta\, \math …

The following question came up when thinking about equidistribution of Satake parameters of elliptic curves. Let $G$ be a compact Lie group with Haar measure $\mathrm{d} x$. Recall that a sequence $\{ …

The paper "λ-equidistributed sequences of partitions and a theorem of the De Bruijn–Post type", by Chersi and Volčič, proves that if $(X,d,\lambda)$ is a separable metric space with probability measur …

Let $G$ be a compact Lie group, $G^\natural$ the space of conjugacy classes in $G$ with the natural pushforward of $G$'s Haar measure. Let $f\in L^2(G^\natural)$. Then the Peter–Weyl Theorem tells us …

When students are first learning about groups, a classic example of a group that is not defined as a set of functions is the group whose underlying set is $\mathbb{R}\setminus-1$, and whose operation …

(Cross-listed from the math stackexchange) Let $(X_i)_{i=1}^n$ be iid random variables with joint cdf $F$. Recall that the empirical distribution function is: $$ F_n(x) = \frac{1}{n} \sum_{i=1}^n \ch …