Solution
= Solution
The truncations $g_n=x^{1/2}\mathbf1_{[1/n,1]}$ are measurable, nonnegative, and increase almost everywhere to $g$. As before their integrals agree with the corresponding Riemann integrals, giving
$$
\int g_n=\int_{1/n}^1x^{1/2}\,dx=\frac23(1-n^{-3/2}).
$$
Monotone convergence therefore gives the finite value $\boxed{\int g=2/3}$. Nonnegativity makes this the integral of the absolute value as well, proving <Lebesgue integrability>.