Show that f is Lebesgue integrable on R?

Question

Show that $f(x)=1/x^{1/2}, x \in (0,1]$ is Lebesgue integrable?

Answer 1

The functions $f_n(x)=\frac1{x^{1/2}}\,1_{(1/n,1]}(x)$ are non-negative, bounded, and measurable. Moreover, $f_n\nearrow f$. By monotone convergence, $$ \int_{(0,1]}f=\lim_n\int_{(0,1]}f_n=\lim_n\int_{1/n}^1\frac{dx}{x^{1/2}}=\lim_n\,2\left(1-\frac1{\sqrt n}\right)=2 $$