Forgive my notation! I don't know much about measure theoretic probability theory.
From undergrad probability I learned that a marginal density can be obtained from a joint density function using $$f(x) = \int_{\mathbb{R}} f(x,y) dy$$ for R.V. $X,Y \in \mathbb{R}$.
Is there something similar with probability measures? Something akin to $$\mathbb{P}_x = \int_Y \mathbb{P}_{x,y} d\lambda(y)$$ where $y \in Y$? Does this even make sense?
I looked around and couldn't find an exact answer to this online. Thank you!
Say $(X, Y)$ is a random variable on $\mathbb{R}^2$. By definition, $$P(X \in A) = \int \int 1_A(x)P(x \in dx, y \in dy).$$ Thus we recovered the marginal distribution $P_X$ from the joint distribution $P_{(X, Y)}$. In the case where $(X, Y)$ has a density $f$, we get $P(X \in dx, Y \in dy) = f(x, y)dxdy$, and we recover your formula for the density of $X$. The observation above is not deep. But using conditional probability, you can make further analogs with the continuous case, and these are very deep. See the concept of conditional expectations and regular conditional distributions.