I'm looking for a function f which is not Lebesgue integrable but |f| is integrable or can we say that such function does not exist?
2026-04-02 11:10:21.1775128221
A function whose absolute value is Lebesgue integrable
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Let $A\subset [0,1]$ be a non-measurable set. Then $[0,1]-A$ is also non-measurable. Define $$f(x)=\chi_A-\chi_{[0,1]-A}$$ Then you may verify that $f$ satisfies the condition you desire.