# Expectation of non-negative random variable and answer veryfication

Let $$X$$ be a non-negative random variable then $$\mathbb{E}(|X|) <\infty$$ if and only if $$\sum_{n}^{\infty}\mathbb{P}(|X| > \epsilon n) < \infty$$ for every $$\epsilon > 0$$.

I tried in the following way,

let $$\epsilon = 1$$

Now $$\mathbb{E}(|X|) = \int_0^{\infty}\mathbb{P}(|X|>t)dt\\ = \sum_{n=1}^{\infty} \int_{n-1}^{n}\mathbb{P}(|X|>t)dt\\$$ Since $$\mathbb{P}(|X|>t)$$ is non-increasing function then $$\sum_{n=1}^{\infty}\mathbb{P}(|X|>n)\leq\mathbb{E}(|X|)\leq \sum_{n=1}^{\infty}\mathbb{P}(|X|>n-1)\\ \Rightarrow \sum_{n=1}^{\infty}\mathbb{P}(|X|>n)\leq\mathbb{E}(|X|)\leq 1 +\sum_{n=1}^{\infty}\mathbb{P}(|X|>n)$$ Hence the result follow by comparison test. How can I prove this for every $$\epsilon >0$$.

What you have done is correct if you repalce $$i$$ by $$n$$ throughout.
Note that $$E|X|<\infty$$ iff $$E|\frac X {\epsilon} | <\infty$$ for any $$\epsilon >0$$. Apply what you have proved to the random variable $$\frac X {\epsilon}$$.