Past exam of the options I course of the University of Oxford / 2007 / D / D1 / iii

The truncations $g_n=x^{1/2}{1}_{[1/n,1]}$ are measurable, nonnegative, and increase almost everywhere to $g$. As before their integrals agree with the corresponding Riemann integrals, givingMonotone 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.