I am looking at to two different ways to define $\beta$ mixing coefficients and I am struggling to understand how and why they are the same.
R.Bradley defines:
$$\beta(\mathcal{A, B}) := \sup\frac{1}{2}\sum^{I}_{i=1}\sum^{J}_{j=1}\left|P(A_i \cap B_j) - P(A_i)P(B_j\right)|$$
Paul Doukhan defines: $\beta(a) := ||\mathbb{P}_{-\infty:0}\otimes\mathbb{P}_{a:\infty}-\mathbb{P}_{-\infty:0, a:\infty}||_{TV}$
Could you, please, help me understand how these definitions are equal?