If we have a function defined from $[a,b]$ to the complex numbers, does that function being Lebesgue measurable imply it is also integrable? (a member of $L^1$?)
2026-04-25 02:34:07.1777084447
Does a function being Lebesgue measurable imply it is integrable? ($f\colon [a,b]\to\Bbb C$)
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Note that $f(x)=\begin{cases}0&\text{if $x=a$}\\\frac1{x-a}&\text{otherwise}\end{cases}$ is measurable but not integrable (the integral "should" involve $\ln 0$).