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Is the square root of a Lebesgue integrable function always integrable?

Thanks!

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    Ouch, silly question :) Thanks everyone!2012-02-25

2 Answers 2

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No, with the usual definition that "Lebesgue integrable" means $\int |f|<\infty$. Just take $ f(x)=\frac1{x^2}\,1_{[1,\infty)}. $ Then $ \int_{\mathbb{R}}f=\int_1^\infty\frac1{x^2}=1, $ but $\sqrt{f}=\frac1x\,1_{[1,\infty)}$, and so $ \int_{\mathbb{R}}\sqrt{f}=\infty $

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Consider $f(x)=\frac{1}{x^2}$ on $[1,\infty)$.