A real-valued measurable functionfis said to be in $\displaystyle L^p$-space if $\displaystyle |f|^p$ has finite integral. So if $\displaystyle f\in L^p$, $\displaystyle |f|^p$ must be Lebesgue integrable. My question is: in this case, is it possible thatfitself is not Lebesgue integrable? Under what condition can we have that anfin $\displaystyle L^p$ is also Lebesgue integrable? Thanks!