expectations in poisson point process

Question

here, $\Phi_e$ is a poisson point process and $\eta_k$ a random variable having exponential distribution.

I'm having trouble in understanding how this equality holds?

Answer 1

A way to understand the link between probability and expectation is the following:

$\begin{align} \mathbb P (\max_{e_k\in\Phi_k}(\eta_k)<x) &= \mathbb E_{\Phi_k} ( \textbf 1 \{ \max_{e_k\in\Phi_k}(\eta_k)<x \}) \\ & = \mathbb E_{\Phi_k} ( \textbf 1 \{ \bigcap_{e_k\in\Phi_k}\eta_k<x \}) \\ & = \mathbb E_{\Phi_k} ( \prod_{e_k\in\Phi_k} \textbf 1\{\eta_k<x \}) \\ & = \mathbb E_{\Phi_k}\bigg( \prod_{e_k\in\Phi_k} \mathbb P (\eta_k<x )\bigg) \end{align}$

I only use the independence of the $\eta_k$ (line 2)