Prove the 1-norm of a sequence of $L^1$ functions does not converge

lebesgue-integralreal-analysis

I've been learning Lebesgue Integral recently and the following is a problem in one of our previous sample exams:

If {$f_n$} is a bounded sequence of functions in $L^1(\mathbb{R})$, $f_n\to f$ a.e. and
suppose that for any $M>0$, there exists $N\in\mathbb{N}$ such that $$\int_{\mathbb{R}\backslash[-M,M]}|f_N|>1$$ Prove that $||f_n||_1$ does NOT converge to $||f||_1$.

I tried to separate the integral $||f_n||_1$ into two parts: $[-M,M]$ and $\mathbb{R}\backslash[-M,M]$. The first part converges due to the Bounded Convergence Theorem. So it suffices to prove the second part does NOT converge. But I don't know where to go from there.

I also thought of functions, e.g. $\chi_{[-n-1,-n]\cup[n,n+1]}$ that satisfy the given conditions but had no idea how to use the properties of those functions. Any hint will be much appreciated!

Best Answer

  • First of all, we can notice, as pointed out by PhoemueX, that if $f_n\to f$ almost everywhere and $\lVert f_n\rVert_1\to\lVert f\rVert_1$, then $\lVert f_n-f\rVert_1\to 0$. Consequently, it suffices to prove that we do not have $\lVert f_n-f\rVert_1\to 0$.
  • To do so, it suffices to find a increasing sequence of integers $(n_k)_{k\geqslant 1}$ such that $\lVert f_{n_k}-f\rVert_1\geqslant 1/2$ for each $k$.
  • Let us construct this sequence. Fatou's lemma combined with boundedness in $L^1$ of $(f_n)$ imply that $f$ is integrable. As a consequence, there exists $M_1$ such that $\int_{\mathbb R\setminus [-M_1,M_1]}\lvert f\rvert\leqslant 1/2$.
  • Let $n_1$ be such that $\int_{\mathbb R\setminus [-M_1,M_1]}\lvert f_{n_1}\rvert\gt 1$. Then $\int_{\mathbb R\setminus [-M_1,M_1]}\lvert f_{n_1}-f\rvert\geqslant 1/2$.
  • Suppose now that $n_1<\dots<n_k$ and $M_1<\dots<M_k$ are such that $\int_{\mathbb R\setminus [-M_i,M_i]}\lvert f_{n_i}-f\rvert\geqslant 1/2$ for each $i\in\{1,\dots,k\}$. Let us find $M_{k+1}>M_k$ and $n_{k+1}>n_k$ such that $\int_{\mathbb R\setminus [-M_{k+1},M_{k+1}]}\lvert f_{n_{k+1}}-f\rvert\geqslant 1/2$. Pick $M_{k+1}>M_k$ such that for each $\ell\in\{1,\dots,n_k\}$, $\int_{\mathbb R\setminus [-M_{k+1},M_{k+1}]}\lvert f_\ell\rvert\leqslant 1$. Applying the assumption with $M=M_{k+1}$, we can find $N$ such that $\int_{\mathbb R\setminus [-M_{k+1},M_{k+1}]}\lvert f_N\rvert\gt 1$ and by construction, this $N$ is necessarily bigger than $n_k$, hence $n_{k+1}=N$ does the job.
  • We thus found sequences $(M_k)_{k\geqslant 1}$ and $(n_k)_{k\geqslant 1}$ for which the inequality $\int_{\mathbb R\setminus [-M_k,M_k]}\lvert f_{n_k}-f\rvert\geqslant 1/2$ holds for each $k$. Then $$ \int_{\mathbb R}\lvert f_{n_k}-f\rvert\geqslant \int_{\mathbb R\setminus [-M_k,M_k]}\lvert f_{n_k}-f\rvert\geqslant 1/2 $$ allows to conclude.
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