Is the square root of a Lebesgue integrable function always integrable?
Thanks!
Is the square root of a Lebesgue integrable function always integrable?
Thanks!
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 $
Consider $f(x)=\frac{1}{x^2}$ on $[1,\infty)$.