If $f_n$ is Lebesgue integrable and $f_{n}$ converges pointwise to $f$ then is $f$ Lebesgue integrable?

Question

If $f_n$ is Lebesgue integrable and $f_{n}$ converges pointwise to $f$ then is $f$ Lebesgue integrable?

I know that this is false unless $f_{n}$ converges uniformly to $f$, but is there an example that shows this?

Answer 1

The sequence $f_n(x) = 1/x \cdot \chi_{[1/n,1]}\;(x)\;\;$ is in $L^1([0,1])\;$, and converges pointwise to $1/x \cdot \chi_{]0,1]}\;(x)\;\;$ which is not in $L^1([0,1])\;$ (here $\chi_A\;$ denotes the characteristic function of a measurable subset $A \subset [0,1].$)

A pointwise limit of a sequence of measurable functions is measurable. Here you can't replace "measurable" by "integrable".