Let $Y$ be real valued random variable on probability space $(\Omega, \mathcal{F}, P)$, such that $Y>0$ almost surely. Suppose $(X^a: a\in \Lambda)$ be a set of random variables in the same probability space with the same distribution as $Y$.

[Q.] Is the following true? $$\inf_a (X^a) >0, \quad a.s.-P$$