Probability distribution and "inclusion" Is there such thing as a probability distribution "included" in another?
For example, take two random variables X and Y, where Y takes it values in a set Sy included in Sx. How do you formalize this?
In practice, given observations on variables X and Y, can you test for such an inclusion?

The concept of absolute continuity might interest you. Suppose $X, Y$ are random elements taking values in a space $\mathcal{X}$, with distributions $\mu$ and $\nu$ respectively (these are probability measures on $\mathcal{X}$). Then the inclusion you describe corresponds to $\nu$ being absolutely continuous with respect to $\mu$. That is, for any measurable subset $A\subset \mathcal{X}$ such that $\mu(A)=0$, we also have $\nu(A)=0.$ Or in other words: if $\mathbb{P}(X\in A)=0$, then also $\mathbb{P}(Y\in A)=0.$
This intuitively means that if $X$ cannot take value in some (measurable) subset $A\subset \mathcal{X}$, then $Y$ cannot take value in that subset either.
I know of no straightforward way to test this in general.
