Reference for independent events for general finite measures Two events $A$ and $B$ are independent if $P(A \cap B) = P(A) P(B)$. More generally, for a finite measure $\mu$, it seems that the correct definition of independence is $\mu(\Omega) \mu(A \cap B) = \mu(A) \mu (B)$, where $\Omega$ is the sample space.
One argument for the correctness of my definition is that for the usual probability measure, every event is independent of the sample space. Indeed $P(A \cap \Omega) = P(A) = P(A) \times 1 = P(A) P(\Omega)$. To find the constant $C$ such that $C \mu(A \cap B) = \mu(A) \mu(B)$ defines independence, set $B = \Omega$, then $C \mu(A \cap \Omega) = \mu(A) \mu(\Omega)$ or $C \mu(A) = \mu(A) \mu(\Omega)$, so $C = \mu(\Omega)$, as long as some event has positive measure.
Is there a reference for the definition of independent events for general measures that are not necessarily probability measures?
 A: Your approach is equivalent to normalizing the measure to have total measure 1:
For any set $C$, let $\mu'(C) = \mu(C) / \mu(\Omega)$. Sets $A, B$ are (in the classical sense) independent iff
\begin{align*}
& \mu'(A \cap B) = \mu'(A) \mu'(B) \\
\iff & \frac{\mu(A \cap B)}{\mu(\Omega)} = \frac{\mu(A)}{\mu(\Omega)} \frac{\mu(B)}{\mu(\Omega)} \\
\iff & \mu(\Omega) \mu(A \cap B) = \mu(A) \mu(B)
\end{align*}
as you proposed.
The potential weeakness that I see is that perhaps this notion of "independence" would be nice to define for sets $\Omega$ with $\mu(\Omega) = \infty$, such as $\mathbb R$ with Borel measure. That said, I don't know why one would want to be able to define independence in such contexts; I confess that I'm also not sure why one would want to define independence for any non-probability measure at all.
I suspect that there isn't a commonly-adopted definition that fits general (finite-measure, non-probability) spaces, but if you want one that extends the known one, your definition is probably the right one to use.
