\b[False.] Let $k$ be the indicator of the positive integers. It is measurable, bounded by one and zero <almost everywhere>, so its <Lebesgue integral> on $(0,\infty)$ is zero. Yet $k(n)=1$ for every positive integer $n$, so it does not tend to zero at infinity. <Lebesgue integrability> is unchanged by modifications on a <null set>, whereas a pointwise limit can change.

Solution

Past exam of the Probability, statistics and options II course of the University of Oxford / 2011 / D / D3 / a / ii / Solution

 Solution (source code)