Let $(X, \Sigma,\mu)$ be a $\sigma$-finite measure space, and let $f:X\to\mathbb R^+$ be a measurable function such that $\mu(\{x\in X\,\colon\,f(x)>t\})>\frac{1}{1+t},\; \forall t>0.$ Prove then that $f$ is not integrable.
I've tried to derive a contradiction, however my original plan to use Chebyshev inequality to get such an absurd turned out to be useless..
Can you help me?
Thanks in advance, Guido