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

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SUMMARY

The discussion presents a clear example of a sequence of integrable functions, specifically defined as \( f_n(x) = n^2 x e^{-nx} \), which converges pointwise to the integrable function \( f(x) = 0 \) but does not converge in the \( L^1 \)-norm. The integral of \( f_n \) over the interval \( (0, \infty) \) equals 1, while the integral of the limit function \( f \) equals 0, demonstrating the lack of \( L^1 \) convergence. Additionally, a canonical example is provided involving triangular functions that converge pointwise to zero while maintaining a constant integral value of 1.

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


I think the canonical example here is taking your [itex]f_n[/itex] 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 [itex]x\in \mathbb R[/itex], and I will make [itex]f_n(x) = 0[/itex] for all [itex]n[/itex] sufficiently large), but [itex]\int |f_n| = \int f_n = 1 \neq \int f = 0[/itex].
 
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