I have simple questions on interpreting a joint probability distribution as a measure. Suppose we have a following joint distribution:
$ P(X=x, Y=y) $
where $\sum_{x \in X}\sum_{y \in Y}P(x, y)=1$. Then,
1) Given a specific value of $Y$, e.g., $y_1$, then can $P(X=x, Y=y_1)$ over all $X$ be a positive measure? Can you give me some guidelines for this?
2) Suppose $Y$ consists of two values $y_1$ and $y_2$. Then, is it possible to define a signed measure as follows?
$ \mu(X) = P(X, Y=y_1)-P(X,Y=y_2) $
If $\mu$ is a finite measure so is $A \to \mu (A\cap B)$ for any fixed measurable set $B$; If $\mu$ and $\nu$ are finite positive measures than $ \mu -\nu$ is a signed measure. So the answer to both of your questions is YES.