Can we define probability of an event involving an infinite number of random variables? Consider a collection $(X_a)_{a\in[0,1]}$ of i.i.d. random variables following the uniform distribution on [0,1]. That is, for every real number $a \in [0,1]$ we have a random variable $X_a$. Can we meaningfully define the probability of the event "at least one of the variables $X_a$ equals $1/2$"? And if so, what is that probability?
 A: Hi following my comment I think the answer is no for the cylindrical $\sigma$-algebra $\mathcal{C}$ and  probability measure implied by finite dimensional laws of $X_a$'s. 
Using the lemma in this post :
If $A=\{\exists a\in [0,1]$ $s.t. X_a=1/2\}\in \mathcal{C}$ then there would exist a countable set $S\in [0,1]$ caracterizing $A$. Meaning that for two  elements of $\Omega=\mathbb{R}^{[0,1]}$, $\omega$ and $\omega'$ with $\omega \in A$ such that $X_s(\omega)= X_s(\omega')$, $\forall s\in S$, then $\omega' \in A$ too. 
So let's take $\omega \in A$ equal to $0$ except for one point $a$ not in $S$ where $X_a=1/2$ so that $\omega$ is really in $A$. Now by our lemma, this would imply that $\omega'$ defined by $X_t=0$ for all $t\in [0,1]$ would be in $A$ too but this is false, implying that $A$ is a non-measurable event for the cylindrical $\sigma$-algebra $\mathcal{C}$.
Best regards
A: Assuming we have a countable collection of r.v.'s, $0 \leq P(\cup_i X_i=.5)\leq \sum_1^\infty P(X_i=.5) \leq \sum_1^\infty 0 = 0$. If the collection is infinite, then I feel confident some thing weird will happen.
