2
$\begingroup$

Let $X$ be a random variable in the probability space $(\Omega, F, P)$ such that for all $\epsilon > 0$ there exists a $c > 0$ such that $$ \int\limits_{|X| > c} |X| dP < \epsilon$$ Prove that $X$ is integrable.

I attempted to use the dominated converange theorem, which states like this in my textbook

Let $Y, X_1, X_2,...$ be random variables such that $|X_n| < Y$ for all $n$ and $EY < \infty$. If $X_n \to X$ then $X$ is integrable.

but building a sequence $X_n$ dominated by some random integrable variable that converges to $X$ is something I don't have any insight to do.

Please give me a hint. Anything is greatly appreciated. Thank you.

$\endgroup$
1
$\begingroup$

Hint: $$\left\lvert X\right\rvert=\left\lvert X\right\rvert\mathbf 1\left\{\left\lvert X\right\rvert\leqslant c\right\}+\left\lvert X\right\rvert\mathbf 1\left\{\left\lvert X\right\rvert\gt c\right\}\leqslant c+\left\lvert X\right\rvert\mathbf 1\left\{\left\lvert X\right\rvert\gt c\right\}.$$ Apply this to $c$ such that (for example) $\int\limits_{|X| > c} |X| dP < 1$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.