I want to show that the law of iterated expectations $E[E[X|Y]] = E[X]$ holds for RVs that are not discretely or continuously distributed.
In specific in our class we have defined the expectation of X in terms of the Riemann-Stieltjes integral:
$E[X] = \int_{\mathbb{R}} x d F_X$
What I got so far:
E[E[X|Y]] = $\int_{\mathbb{R}} \int_{\mathbb{R}} x dF_{X|Y} dF_Y$
In the discrete and continuous case we use the definition of the conditional expectation $P(X|Y) P(Y) = P(X\cap Y)$.
My idea was to use the above definition to to define a product measure $F_{X|Y} F_Y = F_{X,Y} = F_{Y|X} F_X$ and apply Fubini's Theorem twice.
\begin{align} \int_{\mathbb{R}} \int_{\mathbb{R}} x dF_{X|Y} dF_Y &= \int_{\mathbb{R}^2} x dF_{X,Y} = \int_{\mathbb{R}} \int_{\mathbb{R}} x dF_{Y|X} dF_X\\ &= \int_{\mathbb{R}} x dF_X \int_{\mathbb{R}} 1 dF_{Y|X} = \int_{\mathbb{R}} x dF_X = E[X] \end{align}
Using the fact that any probability measure integrates to one. Is this a valid application of Fubini's Theorem?
I know that we use the Riemann-Stieltjes Integral to integrate over function with non-continuous CDFs. Can I generally think about the measure $dF(t)$ being equivalent to $dP(X\leq t)$?
Any help or suggestions for introductory readings is appreciated.