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Suppose we have a probability measure $$mu: mathcalS times mathbbRrightarrow [0,1],$$where $mathcalS$ is some countable set. In some personal, handwritten lecture notes I'm reading it is then stated that we "take $(A,B)$ such that $(A,B) sim mu $ and consider $mathbbE[B]$".

I can't parse this sentence. So far, I've come across the "$sim$" notation only where one side was a random variable and the other a density function. But in this case that does not seem to apply.

So I'm not sure what $B$ is; it seems to be some kind of projection of $mu$ onto $mathbbR$. How can I define $B$ precisely/rigorously?

Normally when you have a probability measure $nu$ on some measure space $(mathcalS,mathcalF,nu)$ and you write "let $X$ be a random variable with $Xsim nu$", then it just means that $X$ is a measurable mapping from some probability space $(Omega,mathcalH,P)$ such that $X(P)=nu$. In other words $nu$ is the distribution of $X$.

In your case you are given a probability measure $mu$ on the product space $mathcalStimes mathbbR$. When we now say that $(A,B) sim mu$, we simply mean that $A,B$ are random variables/elements defined on some common probability space $(Omega,mathcalH,P)$ such that the pushforward measure $(A,B)(P)=mu$, that is $mu$ is the simultaneous distribution of $A$ and $B$. Here $(A,B)(P)$ is the pushforward measure of $P$ with the bundle map $omegamapsto(A(omega),B(omega))inmathcalStimesmathbbR$. Constructing the bundle $(A,B)$ like this, will yield that the marginal distributions are given by $Asim pi_1(mu)$ and $B sim pi_2(mu)$

Similarly we can arrive at the following representation of the expectation of $B$:
beginalign*
E(B) &= int_Omega B(omega) , dP(omega) \
&= int_Omega pi_2(A(omega),B(omega)) , dP(omega)\
&=int_mathcalStimes mathbbR pi_2(a,b) , d(A,B)(P)(a,b)\
&= int_mathcalStimes mathbbR pi_2(a,b) , dmu(a,b) \
&= int_ mathbbR b , d pi_2(mu)(b)
endalign*
where we used the abstract change of variable theorem, and that $pi_2$ is the coordinate projection onto the second coordinate, that is $pi_2 :mathcalStimes mathbbRto mathbbR$ given by $pi_2(s,x) =x$ for any $(s,x)inmathcalStimes mathbbR$.

One way to construct $A$ and $B$ rigorously is to let the common probability space be given by $(mathcalStimes mathbbR, mathcalB(mathcal(S))otimes mathcalB(mathbbR),mu)$ and now define the random variables/elements as $A=pi_1$ and $B=pi_2$.

My guess is that $ mu $ is a form of probability mass-distribution function, and "$ (A,B) sim mu $" means that $ A $ and $ B $ are random variables jointly distributed according to $ mu $:
$$ mathrmProbleft( leftA = sright & lefty<Ble xrightright) = muleft(s,xright) - muleft(s,yright) .$$
The distribution function $ F_B $ of $ B $ would then be given by $ F_Bleft(xright) = displaystylesum_sinmathcalSmuleft(s,xright) $, and $ mathbbEleft[Bright] $ by the Lebesgue-Stieljes integral
$$mathbbEleft[Bright] = int_-infty^infty x dF_Bleft(xright) .$$