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.


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) $\int\limits_{|X| > c} |X| dP < 1$.


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