No, have a look at Kolmogorov's 0-1 law
I think considering or the measurability of the rv's doesn't change much thing... I'll have to give a deeper look into the definition of a tail event but I'm quite tired.
Anyway, I think your example doesn't fit what you want...
Let's assume the rv's are independent.
We want to find .
We have .
This condition, added to the independence, lets us use Borel-Cantelli's lemma (part II) and we get that
Now, (for the first equality, we can take because it's dense in )
By , this final probability is 0. Hence .
I agree though that this is kind of counterintuitive. So if there's a mistake somewhere just tell me!