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.