1
$\begingroup$

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

Thanks!

  • 0
    Ouch, silly question :) Thanks everyone!2012-02-25

2 Answers 2

2

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 $$ \int_{\mathbb{R}}\sqrt{f}=\infty $$

2

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