Quote from book: "Consider an arbitrary atomic probability measure $\Gamma$ on unit sphere. Let $(\sigma_{l})$ denote an i.i.d sample from $\Gamma$."
I don't understand the second sentence. Does it mean
Let $(\sigma_{l})$ be random variables mapping into the sphere. Then pick the ones, mapping into the support of $\Gamma$.
or
Pick points $(\sigma_{l})$ uniformly from the support of $\Gamma$.
So that in the case $\Gamma$ is a random measure (i.e. $\Gamma:\Omega\times \mathbb{S}^{n-1}\to \mathbb{R}$), the second sentence respectively says
Let $(\sigma_{l})$ be random variables mapping into the sphere. Then pick the ones, mapping into the most probable support of $\Gamma$.
or
Pick points $\sigma_{l}$ uniformly from the most probable support of $\Gamma$.
Thank you
(1)"The Sherrington-Kirkpatrick model" by D.Panchenko pg. 17
Best Answer
It means that each $\sigma_i$ is a random variable (mapping into the sphere) with distribution given by $\Gamma$. Furthermore, $\sigma_1,\dots, \sigma_n$ are (stochastically) independent for each $n$.