Given a joint distribution $X,Y$, I'm trying to get the probability of $X=x$ using the sum rule: $$P(x)=\int f(x,y)dy.$$ I undestand that I need to solve for the normalizing constant $K$, to obtain a well-defined density function: $$1=\int K f(x)dx, $$ but I've also read that we can have (Zadeh 1968): $$P(x)=\mathbb{E}[f(x)]$$
for any positive, measurable and unnormalized $f$. From this, I'm inferring that $$P(x)=\int K f(x,y)dy=\mathbb{E}_Y[f(x,y)]$$ holds for some set-valued $X$, which I do not know if it is true (and how to proof it, in that case): please help.
Zadeh, L. A., Probability measures of fuzzy events, J. Math. Anal. Appl. 23, 421-427 (1968). ZBL0174.49002.