This is a question for the interchange of a limit and an expectation, in the special case where the random variable of interest is an infinite sum of indicator functions.
Given, $$ \mathbb{E}\left[\lim_{t\rightarrow\infty}\sum_{i=0}^{t}{1}(A_i)\right]=c < \infty $$ where $1(\cdot)$ is the indicator function, $A_i$ is some event. What additional conditions (if any) are required to guarantee that the limit and the expectation operator may be interchanged, such that it holds that $$ \lim_{t\rightarrow\infty}[\sum_{i=0}^{t}\mathbb{P}(A_i)]=c $$
Because the indicator functions are non-negative then you can use the monotone convergence theorem to exchange expectation and summation sign.