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Is there a sequence of functions where $\int_0^1|f_n(x)|->0$ as n approaches infinity, but the sequence of functions is also pointwise divergent over every x in $[0,1]$?

Initially I thought cos(nx) could be an answer but that is not pointwise divergent at x=0. My other solutions also did not satisfy the integral due to the absolute value.

user1513121
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The hypothesis immediately implies that the sequence of functions converges to 0 in the $L_1$ sense. Nevertheless, we can build such a sequence that is pointwise divergent for every point, as you desire.

First, find $A_i$ a sequence of intervals such that:

  • the length of the intervals goes to zero, and
  • every point is in infinitely and coinfinitely many of the intervals.

It is not so hard to find ways to do this.

The sequence $\chi_{A_i}$ of characteristic functions of these intervals has the properties you requested.