Application of Borel Cantelli Lemma

Question

Let X be a non-negative random variable with finite expected value. Can we use Borel Cantelli Lemma to show that X is finite?

Answer 1

This is simpler than the Borel–Cantelli lemma: if $\Pr(X=\infty)>0$, then $\operatorname{E}(X)=\infty$ by the definition of expectation.

Now suppose $X$ is a discrete random variable equal to the number of events in a sequence $A_1,A_2,A_3,\ldots$ that actually occur. Then one could think of applying Borel–Cantelli. One would have to know that the hypotheses are satisfied.

Answer 2

Let $(a_n)$ be a monotone-increasing sequence of non-negative numbers. By the Markov's inequality,

$$ \Bbb{P}(X \geq a_n) \leq \frac{\Bbb{E}X}{a_n}. $$

Choose $a_n$ such that $\sum_n \frac{1}{a_n} < \infty$. Then $\sum_n \Bbb{P}(X \geq a_n) < \infty$ and hence by the Borel-Cantelli's lemma,

$$ \Bbb{P}(X \geq a_n \text{ i.o.}) = 0. $$

But if $X(\omega) = \infty$, then we must have $X(\omega) \geq a_n$ infinitely often. Therefore we have $\Bbb{P}(X = \infty) = 0$.