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

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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 $0$.

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Suppose $Y$ is equal to 1/n with probability $2^{-n}$, and suppose your index set is infinite. Then $Y>0$ always, but your infimum is $0$ with probability 1.

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