Proof of $\mathbb{E}(Y) = \sum_{t = 0}^{\infty} \mathbb{P}(Y > t) $ for the infinite case?

Question

Let $Y$ be a non-negative integer random variable, then show that:

$\mathbb{E}(Y) = \sum_{t = 0}^{\infty} \mathbb{P}(Y > t) $

Attempt:

We can clearly see that $Y = \sum_{t = 0}^{\infty} \mathbf{1}_{Y > t} $.

And by the linearity of expectation for countable sums (see here: Expected value of infinite sum) we can see that if $ \sum_{t = 0}^{\infty} \mathbb{P}(Y > t) < \infty $ , then the above result is shown. But what happens when $ \sum_{t = 0}^{\infty} \mathbb{P}(Y > t) = \infty$, can we somehow show that $\mathbb{E}(Y) = \infty$ in this case, or does this not hold?

Answer 1

The comments have suggested using Tonelli's theorem, but I'll propose a solution using the Monotone Convergence Theorem. Indeed applying the Monotone Convergence Theorem we get that

$$\mathbb{E}[Y]=\mathbb{E}\left[\sum_{j=0}^\infty\mathbf{1}_{Y> j}\right]=\lim_{n\to\infty}\mathbb{E}\left[\sum_{j=0}^n\mathbf{1}_{Y> j}\right]=\lim_{n\to\infty}\sum_{j=0}^n\mathbb{E}[\mathbf{1}_{Y>j}]=\sum_{j=0}^\infty\mathbb{P}(Y>j).$$