Is "ensemble" a standard term in probability theory? The book "information theory, inference, and learning algorithms" uses a definition to formalize probability:

An ensemble $X$ is a triple $(x,\mathcal A_X,\mathcal P_X)$ where the outcome $x$ is the value of a random variable, which takes on one of a set of possible values, $\mathcal A_x =\{a_1,a_2,\dots,a_i,\dots,a_I\}$, having probabilities $\mathcal P_X = \{p_1,p_2,\dots,p_I\}$, with $P(x=a_i)=p_i$, $p_i\ge0$ and $\sum\limits_{a_i\in\mathcal A_X} P(x=a_i)=1$.

I haven't seen this definition before (I am used to the concept of a probability space). Is this a standard definition? Does it relate to "ensemble" in statistical physics in any way? I haven't found much on it online.
 A: Since the realization of a random variable is formally nothing but the random variable itself (or a copy of it), the given definition is just the definition of a discrete random variable. I had a look at the book and the term ensemble is actually used synonymously to random variable.
The author himself claims: "[...] we will sometimes need to be careful to distinguish between a random variable, the value of the random variable, and the proposition that asserts that the random variable has a particular value". I see that this might simplify one's thinking in certain situations, but mathematically there is no reason to distinguish between a random variable and its value (and also there is no possibility to do so, since the definitions agree on a formal level).
In several areas of statistics and numerical analysis, the term "ensemble" refers to a large number of realizations $x_1,\dots,x_n\sim x$ (often, but not always, independent), such as Monte Carlo methods and similar. A very typical application are stochastic processes, such as particle methods or ensemble forecasting, where these realizations describe the statistical behavior of the process for large $n$. This is then very much related to statistical physics, where one might think of the realizations being gas particles moving according to some (deterministic or stochastic) law.
But this notion of ensemble goes beyond the one used in MacKay's book, as explained above.
