Consider two random variables $X$ and $Y$, with $X$ discrete and $Y$ could be continuous or discrete.
Assume $0<Pr(X=0)=E(Pr(X=0|Y))$ where the equality follows from the law of total probability.
As $E(Pr(X=0|Y))>0$, it means that $Pr(X=0|Y)=0$ a.s. is impossible.
Question: could you clarify what is the meaning of $Pr(X=0|Y)=0$ almost surely?
My thoughts: following here, $Pr(X=0|Y)=0$ almost surely means that the set of values $y$ of $Y$ such that $Pr(X=0|Y=y)>0$ has probability zero. However, I am confused when $Y$ is a continuous random variable because the probability of $Y$ taking a specific value is always zero.
Could you help me to clarify this point?
Let $$ S = \{ y \in Y : {\rm Pr}(X = 0 | y ) = 0 \}.$$ So $S$ is the set of $Y$-values for which there is a zero conditional probability of $X$ being zero.
If we say that ${\rm Pr}(X = 0 | y) = 0$ almost surely, this means that $Y \ \backslash \ S$ is a null subset of $Y$. In other words, if you pick a $Y$-value at random, then the probability of picking a value $y$ for which ${\rm Pr}(X = 0 | y) = 0$ is a probability of one.
In your situation, you know that $\mathbb E ({\rm Pr}(X = 0 | y) ) > 0 $. Hence the subset of $Y$ on which ${\rm Pr}(X = 0 | y ) > 0$ must have strictly positive measure. But this set is precisely $Y \backslash S$. So $Y \backslash S$ is not a null set, i.e. it is not the case the ${\rm Pr}(X = 0 | y)$ almost surely.