Let $\mathcal E=\{\pm e_j\mid j\in \{1,...,d\}\}$ where $e_j\in\mathbb Z^d$ is such that $e_j=(0,...,0,1,0,...,0)$ where the $1$ is at the $j-$th position. In an article, it's written : " Let $\mathcal M_{\mathcal E}$ the set of probability measures on $\mathcal E$, i.e. vectors with $2d$ non negative-entries with sum up to 1".
I don't really understand what it mean... is it vectors of the form $(x_1,...,x_d,...x_{2d})$ s.t. $\sum_{i=1}^{2d}x_i=1$ ?
I tried to understand with $d=2$. So I have $\mathcal E=\{(1,0),(0,1),(-1,0),(0,-1)\}$. What would be $\mathcal M_{\mathcal E}$ ?
There's no actual measure theory in this discrete probability question- the author is really needlessly complicating things.
Yes, although the probabilities also have to be nonnegative.
For $d=2$,
$M_{E}=\left\{ x \in R^{4} | \; x_{1}+x_{2}+x_{3}+x_{4}=1, x\geq 0 \right\}$