I am wondering whether someone can help me derive this equation in Billingsley (1968), Equation (20.49)? Suppose $u_i$ and $v_j$ are integers with $u_1 \leq v_1 < u_2 \leq v_2 < \dots < u_r \leq v_r$ and $u_i - v_{i-1} \geq b,$ $i=2,\dots,r$, and suppose that, for $I=1,\dots,r$ $E_i$ lies in $\mathcal{M}^{v_i}_{u_i},$ where $\mathcal{M}$ is a relevant sigma field.
The text says that it follows from the definition of "$\varphi$-mixing by induction on $r$."
$$|P ( E_1 \cap E_2 \cap \dots \cap E_r) - PE_1 PE_2 \dots PE_r | \leq r \varphi(b).$$
Can someone help me derive this relationship?
I tried deriving this from the $\varphi$ mixing condition $|P(E_1 \cap E_2) - P E_1 P E_2| \leq \varphi (n) P E_1 $ but somehow could not find a way to do it. One thing I tried was to write this for three events to see if it factors:
$P(E_1\cap E_2 \cap E_3) = P(E_1 | E_2 \cap E_3) P(E_2 | E_3) P(E_3)$ which could be inserted into the top equation for $r=3$ but it somehow does not seem to reduce to the basic definition.
Any help would be much appreciated.
Suppose that we have shown the assertion for $r$ and let us show it for $r+1$. Take $E_i\in\mathcal M_{u_i}^{v_i}$ with $u_1 \leq v_1 < u_2 \leq v_2 < \dots < u_r \leq v_r<u_{r+1}\leq v_{r+1}$ and $u_{r+1}-v_r\geqslant b$. We can assume that $F=\bigcap_{i=1}^r E_i$ has a positive probability, otherwise there is nothing to prove. The key point is to write $\mathbb P(F\cap E_{r+1})$ as a conditional probability, making appear a quantity involved in the definition of $\varphi$-mixing coefficients: \begin{align} \left\lvert \mathbb P\left(\bigcap_{i=1}^{r+1}E_i\right)-\prod_{i=1}^{r+1}\mathbb P(E_i)\right\rvert&=\left\lvert \mathbb P\left(F\cap E_{r+1}\right)-\prod_{i=1}^{r+1}\mathbb P(E_i)\right\rvert\\ &=\left\lvert \mathbb P\left( E_{r+1}\mid F\right)\mathbb P(F)-\prod_{i=1}^{r+1}\mathbb P(E_i)\right\rvert\\ &=\left\lvert \left(\mathbb P\left( E_{r+1}\mid F\right)-\mathbb P\left(E_{r+1}\right)\right)\mathbb P(F)+\mathbb P(F)\mathbb P(E_{r+1})-\prod_{i=1}^{r+1}\mathbb P(E_i)\right\rvert\\ &\leqslant \left\lvert \mathbb P\left( E_{r+1}\mid F\right)-\mathbb P\left(E_{r+1}\right) \right\rvert \mathbb P(F)+\mathbb P(E_{r+1})\left\lvert \mathbb P(F)-\prod_{i=1}^{r}\mathbb P(E_i)\right\rvert. \end{align} The first term is smaller than $\varphi(b)$ by definition and the second one than $r\varphi(b)$ by the induction assumption.