I want to give an example of a sequence of functions $f_1 \dots f_n$ that converges with respect to the metric $d(f,g) = \int_a^b |f(x) – g(x)| dx$ but does not converge pointwise.
I'm thinking of a function $f_n$ that is piecewise triangle, whose area converges to some constant function, but doesn't converge pointwise.
I just can't manage to formalize it.
Best Answer
Here's an example of a sequence, mentioned by Sam in the comments, of a sequence that converges pointwise nowhere on $[0,1]$ but $\int_{[0,1]} f_n\rightarrow 0$:
Let
$\ \ f_1=\chi_{[0,1]}$, $f_2=\chi_{[0,{1\over2}]}$, $f_3=\chi_{[{1\over2},1]}$, $f_4=\chi_{[0,{1\over4}]}$, $f_5=\chi_{[{1\over4},{2\over4}]}$, $f_6=\chi_{[{2\over4},{3\over4}]}$, $f_7=\chi_{[{3\over4},1]}$, $\ldots$.
Here $\chi_A$ is the function whose value is 1 on the set $A$ and $0$ on the set $A^C$.