Past exam of the options II course of the University of Oxford / 2013 / D4 / e

For each integer $m\ge 1$, list the $m^2$ functionsin increasing $j$, and concatenate the blocks in increasing $m$. Each function has $L^1$ norm $1/m$, so the resulting sequence has norm tending to zero. Every $x\in (0,1)$ lies in exactly one interval of each block, where the function takes value $m$. ThereforeThis unbounded typewriter sequence with vanishing L1 norm is a moving-spike construction: the support shrinks fast enough to beat the increasing height.

No such sequence exists. By Fatou's lemma on $(0,1)$,But the proposed nonnegative measurable integrand is positive at every point of $(0,1)$, so its integral cannot be zero. Explicitly, the sets where it is at least $1/k$ cover $(0,1)$; at least one has positive measure, forcing a positive integral. This contradicts the displayed bound.