Having trouble understanding the concept of "mixing" in dynamical systems. I'm trying to understand the concept of mixing in dynamical systems theory, especially when the system in question has a measure-preserving flow. Here's how the condition is expressed mathematically: If $\mu$ is the measure and $\phi$ is the flow, then for all subsets $A$ and $B$ of positive measure, $\lim_{t \rightarrow \infty}\mu(\phi^{t}(B) \cap A) = \mu(B) \times \mu(A)$.
Now suppose $B$ is an arbitrary set with measure greater than 0 and less than 1. If the flow is measure preserving, then for all $t$, $\mu(\phi^{t}(B)) = \mu(B)$. Pick $A = \lim_{t \rightarrow \infty}\phi^{t}(B)$. Then, $\mu(A) = \mu(B)$. It follows that $\lim_{t \rightarrow \infty}\mu(\phi^{t}(B) \cap A) = \mu(A) = \mu(B)$. 
So if the dynamics is mixing, then we will have $\mu(B) = \mu(B) \times \mu(B)$. But this is only possible if $\mu(B)$ is 0 or 1, contradicting our initial assumption.
Isn't this a problem with the definition of mixing? Is the definition in my source wrong? Or am I doing something wrong?
 A: There seems to be suspicious reasoning as pointed out in the comments when you define $A$ and then conclude that since $\mu(A)=\mu(B)$ then $\lim_{t\rightarrow\infty}\mu(\phi^t(B)\cap A)=\mu(A)$. 
Have you encountered ergodic theory? Your mixing condition is more formally referred to as "strong mixing" which implies ergodicity, in that $\phi_t$ must be ergodic. By definition a flow is ergodic when $A=\phi_{-t}(A)$ implies $\mu(A)=0,1$.  
A: You may want to think about the notion of independent events in probability theory: two events are independent if $Pr(A \cap B) = Pr(A)\cdot Pr(B).$
So the definition you give says that in the large time limit, the events of a point
being in $\phi^t(B)$ and in $A$ are independent. So however $A$ and $B$ are positioned with respect to one another, after a long time $t$, the position of $\phi^t(B)$ is completely independent of the position of $A$.  
Intuitvely, the points in $B$ are being completely mixed throughout the set, independently of where they were originally positioned.  Hence the term mixing.
A: I think the problem is the assumption "Pick $A=\lim_{t \to \infty} \phi^t(B))$" It does not make sense to define the limit of a sequence of sets and expect to get a measurable set $A$. While in the original expression of mixing, only a sequence of non negative numbers are taken limit: $\lim_{t \to \infty} \mu(\phi^t(B) \cap A)$ . Everything inside $\mu()$ is at finite time, and $\mu$ brings them to a sequence of numbers, and numbers are taken limit. 
In other words, you are not allowed to move the $\lim$ into the parenthesis of $\mu()$.
