What is the difference between a statistical ensemble and a probability space?

Question

I am not a physicist, and I'm mainly used to thinking about probability spaces (formulated in measure theory).

I am trying to understand what a "statistical ensemble" is, and how it is different from a probability space in terms of measure theory.

In measure theory, a probability space is simply a tuple $(A,\Sigma, P)$, where $\Sigma\subseteq \mathcal P (A)$, and $P:\Sigma\to [0,1]$. My guess would be that an ensemble is simply a subset $A^*\subseteq A$, where $A$ is interpreted as the state space of a physical system, and $A^*$ is a subset of that state space where we have that a certain function(s) has a fixed value(s). For example, take the total energy of the system $E:A\to [0,\infty]$. We might let $A^*=\{a\in A: E(a)=E^*\}$ for some constant $E^*$. Assuming for a moment that we are dealing with a quantum mechanical system so the state space is finite, we can then define a probability measure $P$ on the measurable space $(A^*, \mathcal P(A^*))$, such that for all $a,b\in A^*$, we have that $P(a)=P(b)$. (This is then the fundamental postulate of statistical mechanics, if I'm not mistaken). Thus in this case, our ensemble would be the probability space $(A^*, \mathcal P (A^*), P)$.

Questions:

1. Is my understanding correct?

2. Is there a more subtle difference between the two?

3. Is there a good reason that "ensemble" is terminologically different from the terms used in probability theory, or is this purely a historical quirk? What would be the equivalent term as used by probability theorists?

Answer 1

From a practical point of view, statistical ensembles are just probability spaces, and the use of two different terminologies is mainly due to historical reasons.

The (general, abstract) notion of probability space was introduced in the 1930s by Kolmogorov in his famous work. (Of course, particular instances of this notion had already been used before.)

The notion of statistical ensembles (which he called monodes, special cases being ergode (=microcanonical) and holode (=canonical)) was introduced earlier by Boltzmann in the 1880s. The modern terminology (microcanonical, canonical, grand canonical) was then introduced by Gibbs some years later (and was of course inspired by Boltzmann).