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Is it true that |X| vanishes at infinity, then X is integrable?

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Suppose $X$ is a random variable on $(\Omega,\mathscr{F},\mathbb{P})$. If there exists $M>0$, such that for all $\lambda>0$, we have $\mathbb{P}[|X|>\lambda]\le\frac{M}{\lambda}$, then is it true that $X$ is integrable?

This condition is weaker than the absolute continuty: Does absolute continuity of integral imply integrability on finite measure space, but compared to absolute continuity, the condition above gives a explicit inequality. So, I'm not sure if it is true.

My attempt:$$\mathbb{E}[|X|]=\int_0^{+\infty}\mathbb{P}(|X|>\lambda)\le M\int_0^{+\infty}\frac{1}{\lambda}d\lambda,$$ but the right side is infinite.


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