Past exam of the options I course of the University of Oxford / 2006 / D / D1 / ii / Solution

A real measurable function is Lebesgue integrable precisely when the integral of its absolute value is finite. For every integer $k\ge 1$, on $I_k=[k+1/6,k+5/6]$ one has $|\sin \pi x|\ge 1/2$ and $x\le k+1$. ConsequentlyThe intervals have disjoint interiors. Countable additivity of the nonnegative Lebesgue integral, or the monotone convergence theorem applied to their finite unions, givesThe harmonic series diverges, so the function is not Lebesgue integrable. Cancellation between adjacent oscillations cannot give absolute integrability.