Linearity of expectation for indicator sums

Write $X = \sum_{i=1}^m \mathbb{1}[A_i]$ where $\mathbb{1}[A_i]$ is $1$ if $A_i$ occurs and $0$ otherwise. Expectation is itself defined as a sum (or integral) over outcomes weighted by their probability, and this sum is always additive over a finite list of random variables — this is true regardless of whether the variables are independent, since additivity of expectation never uses independence, only that we are summing the same underlying probability measure.

Formally, $\mathbb{E}[X] = \mathbb{E}\left[\sum_{i=1}^m \mathbb{1}[A_i]\right] = \sum_{i=1}^m \mathbb{E}[\mathbb{1}[A_i]]$ by additivity of expectation over a finite sum of random variables.

Finally, for any indicator variable, $\mathbb{E}[\mathbb{1}[A_i]] = 1 \cdot \Pr[A_i] + 0 \cdot \Pr[\overline{A_i}] = \Pr[A_i]$. Substituting this into the sum gives exactly $\mathbb{E}[X] = \sum_{i=1}^m \Pr[A_i]$, with no assumption on how the $A_i$ relate to one another.