Sufficient condition for finite Expectation

Question

In the lecture the lecturer wrote:

"Assume that for a random variable $X$ it holds that $X \geq 0$ or $X \in L^1$. Then $\mathbb{E}[X] < \infty$."

I can understand (or partially understand) why integrability implies finiteness of expectation. But why should non-negativity imply that the expectation exists?

Did I got something wrong?

Thank you very much.

Answer 1

Of course non-negativity does not imply finite expected value. Consider discrete distribution: $P(X=k)=\frac{1}{k(k+1)} $ for $k=1,2,...$