A mixed discrete-continuous joint law, checked term by term

Suppose (X, Y) is a two-dimensional random vector with range $(0,\infty)\times\{1,2\}$, such that for any $A\subset(0,\infty)$ and $y=1,2$, $P(X\in A,Y=y)=\tfrac12\int_Aye^{-yx}dx$. Then, which of the following statements is false?