I've been confused by the naming convention used in Meckes' book on Random matrix theory on the classical compact group that I'm hoping someone could help me clarify. The book can be found https://case.edu/artsci/math/esmeckes/Haar_book.pdf
I understand a left Haar-measure is one that is left-invariant. Assuming $G$ is a topological group then for all $A\subseteq G$ and $g \in G$, we say $\mu$ is a left-hair measure if $\mu(gA) = \mu(A)$. But I'm confused to what they mean when they say a random $g \in G$ is Haar-distributed? I presume they mean a random variable $g:\Omega \to G$ for $(\Omega,\mathbb{P})$ some probability space? And it's Haar-distributed to mean for all $h \in G$ and $A\subset G$, $$\mathbb{P}(g \in A) = \mathbb{P}(hg \in A).$$ Honestly, I'm not sure. Also, does Haar-distributed mean the same thing as distributed according to a Haar measure? Would really appreciate the clarification.
For intuition the simplest nontrivial case to think about is a Haar-random element of the circle $S^1$: this means, to be very concrete, picking a random angle $\theta \in [0, 2\pi]$ in such a way that $\mathbb{P}(\alpha < \theta < \beta) = \frac{\beta - \alpha}{2\pi}$. In other words we want to pick a random element of $S^1$ in a "uniformly distributed" way. This generalizes to any compact topological group and is made precise using the definition of Haar measure.
A random Haar-distributed $g \in G$ (which means the same thing as "distributed according to Haar measure") can be defined formally as a random variable, but that random variable is itself valued in $G$, and the function is the identity function $G \to G$. The important content of this statement is that $\mathbb{P}(g \in U) = \mu(U)$ for measurable $U$ and $\mu$ the Haar measure.