I know this is almost trivial or evident, but I wanted to be sure if any Lebesgue integrable function on a compact set is bounded or not. I hope it is true because I'm using that fact for solving some of my problems. If it is not true, please provide a counterexample. Thanks.
2026-05-15 17:02:51.1778864571
if $f$ is Lebesgue integrable in $[0,1] \Rightarrow f$ is bounded in $[0,1]$?
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Unfortunately this is not true: let $f(x)=\frac{1}{\sqrt{x}}$ for $0<x\leq 1$, with $f(0)$ defined however you want. Then $f$ is integrable on $[0,1]$ but not bounded.