What does ‘$P$-almost-surely unique’ mean

Question

I'm reviewing the Radon-Nikodym Theorem. The theorem says:

If $Q\ll P$, then there are exist a integrable random variable $\epsilon$ that is $P$-almost-surely unique, and it is such that for each event $A$:
$$Q(A)=\int_A \epsilon\, dP $$

My question: In this context what mean P-almost-surely unique?

Answer 1

If $\epsilon, \epsilon'$ in your case both satisfy Radon-Nikodym, then $\epsilon = \epsilon'$ up to some $P$-null-set. That is, it only differs by some set $A$, where $P(A)=0$.