Sequence of Integrable Fns Converging to Integrable Fn But Not in L1-Norm

[kikkka]

Dear friends can you show me please an example of a sequence of integrable functions fn:R->R converging to an integrable function f but *not* in the L1-norm, i.e. such that
\Int \mid f_n -f\mid is not equal to 0?
Thank u a lot

 

[mathman]

Let f_n(x)=n²xe^-nx. The integral (0,∞) of f_n(x) = 1, but the point wise limit = 0, so we don't have L1 convergence.

 

[AxiomOfChoice]

I think the canonical example here is taking your f_n to have graphs that are triangles of decreasing width but increasing height that always have one vertex at the origin, so that the pointwise limit is 0 (give me any x\in \mathbb R, and I will make f_n(x) = 0 for all n sufficiently large), but \int |f_n| = \int f_n = 1 \neq \int f = 0.