I have a probability space $(\Omega, M, \mathbb{P})$, where each $\omega \in \Omega$ is a random subset of natural numbers (i.e. This is a probability space of sequence of natural numbers sometimes used in probabilistic number theory. Even though it is referred to as sequences, it is actually subsets).
Suppose I have a collection of triplets of natural numbers $F$. Denote $F(\omega) = \{ \theta = \{ n_1, n_2, n_3 \} \in F : \theta \subseteq \omega \}$. In particular, for each $\omega \in \Omega$, $|F(\omega)|$ counts the number of triplets in $F$ that is a subset of $\omega$, and it is a random variable.
I know that the expectation is defined to be $$ \mathbb{E} (|F(\omega)|) = \int_{\Omega} |F(\omega)| d\mathbb{P}. $$ Could someone please explain me why $$ \mathbb{E}(|F(\omega)|) = \sum_{\theta \in F} \mathbb{P}( \theta \subseteq \omega ) $$ holds true? Thanks!