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Scheffé lemma says that: If $f_n\to f$ pointwise and if $\int|f_n|\to \int|f|$ then $f_n\to f$ in $L^1$.

I have several questions around this result.

Q1) If $f_n\to f$ pointwise and that $f_n\in L^1$ for all $n$ and $f\in L^1$, does $f_n\to f$ in $L^1$ ? In other word, if $f_n\to f$ poitwise and $f_n\in L^1$ and $f\in L^1$ for all $n$, does $\lim_{n\to \infty }\int |f_n|=\int |f|$ ?

Atempts: If could look similar to Dominated convergence theorem, but unfortunately $|f_n(x)|\leq |f(x)|$ doesn't hold. So maybe it doesn't work, but I don't have any counter example.

Q2) Can Scheffé lemma be generalized in $L^p$ ? i.e. if $f_n\to f$ pointwise and $\int |f_n|^p\to \int |f|^p$, does $f_n\to f$ in $L^p$ ?

Attempts: I tried to apply Scheffé lemma to $g_n=|f_n|^p$ and $g=|f|^p$, but I just get that $\lim_{n\to \infty }\int||f_n|^p-|f|^p|=0$, but not $\lim_{n\to \infty }\int |f_n-f|^p=0$.

Dylan
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  1. No, as the example $f_n=c_n\mathbf 1_{(0,1/n)}$ shows: when $c_n=n^2$ or $c_n=n$, the pointwise convergence to $0$ hold and $f_n$ is integrable. In the first case, the sequence of the $\mathbb L^1$-norms is not bounded hence there is no hope for a convergence, and in the second case the sequence of the $\mathbb L^1$-norms is equal to one, but this is not the $\mathbb L^1$-norm of the pointwise limit of $(f_n)$.

  2. There exists indeed a generalization in $\mathbb L^p$.

Davide Giraudo
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