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

False. Take $k(x)=1/(1+|x|)$. This is bounded and measurable on $\mathbb{R}$, and tends to zero as $x\to \infty$, butThus decay to zero alone does not imply Lebesgue integrability.

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.