What is a "mixing probability distribution" 
Given a distribution function $F$ such that $\int |x| dF(x)< \infty$ we introduce the mixing probability measure
$$\nu (du,dw):=\frac{1}{\alpha}(w-u)1_{\{u \le 0<w \}}(u,w)dF(u)dF(w) $$
I have two questions: 1) (loosely speaking) what is a mixing probability measure? and 2) how does $\nu$ evaluate a rectangle $A=[a_1,b_1]\times[a_2,b_2] \in \mathbf{R}$?


For 2) first I'm not sure of how to interpret the notation, but I guess it means $\nu(A) \mapsto \int 1_A(u,w)\frac{1}{\alpha}(w-u)1_{\{u \le 0<w \}}(u,w)dF(u)dF(w)$. If this is correct, can one intuitively say how $\nu$ evaluates a rectangle $[a_1,b_1]\times[a_2,b_2]$ from the measure $P_X(x):=F(x)$? [Looking at it I believe the rectangle gets evaluated according to the product measure $P_X \otimes P_X$ but only the part in the second quadrant, and then we do some re-scaling??]
Most grateful for any help provided!
 A: I am only familiar with mixing as a property of measures related to infinite paths, so some dynamics should be involved - there you would expect sometimes the measure to start behaving like a product one. In your case instead one defines a new measure $\nu$ based on one old measure with a CDF $F$ (let's denote that old measure as $\mu$). For this reason I find it hard to tell, why is this new measure called mixing. That's regarding your first question. Perhaps, it's just authors initiative to do so. Perhaps, it would have helped if you provided more context. 
Regarding the second question: I think you've understood the notation correctly, though I do dislike it: the use of both $\nu(\mathrm du,\mathrm dw)$ and $\mathrm dF(u)$, and who would use $u$ and $w$ if you can use $u$ and $v$? Anyways. What is $\alpha$ here as well? 
As an example of how to compute an integral, let's take a rectangle that makes all indicators go positive. That is, let $b_1 \leq 0 < a_2$. Then
$$
\nu([a_1, b_1]\times [a_2, b_2]) = \frac1\alpha\cdot \int\limits_{[a_1, b_1]\times [a_2, b_2]}(v-u)(\mu\otimes \mu)(\mathrm du\times \mathrm dv) = \frac1\alpha\left(\int_{a_2}^{b_2}v\mu(\mathrm dv) - \int_{a_1}^{b_1}u\mu(\mathrm du)\right).
$$
And other cases are treated similarly. 
