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?
$$\inf\_a (X^a) >0, \quad a.s.-P$$



infimum of  a set of positive r.v. with the same distribution
No, certainly not. Let $Y \sim U(0,1)$, so $Y > 0$ a.s. If $\{X^a : a \in \mathbb{N}\}$ are iid $U(0,1)$, then it is easy to see that $\inf\_a X^a = 0$ a.s. In fact this will be true for any $Y$ with essential infimum $0$.


