$f_n, f \in L^1$ and pointwise convergence implies ...

Question

Let $f_n, f \in L^1(\mathbb{R}), f_n \to f$ pointwise. Then $\int \left|f_n-f\right| \to 0$ iff $\int \left|f_n\right| \to \int \left|f\right|$.

My attempt was to use the Triangle Inequality for one side. Certainly taking convergence in mean as true we have convergence in mean of the absolute values, because of the triangle inequality.

Now, take convergence in mean of the absolute values. How to show convergence in mean of the original functions? I'm not even sure where pointwise convergence comes into play at this point. The only thing I tried was Egoroff's Theorem, and that led to nothing.

I would appreciate a hint, or a link if this is a repeated problem (I have a feeling it's standard, but I couldn't find it in the drop-down menu when I typed out the title).

Your space has finite measure? Which norm is $| f_n - f|$ ? – Rodrigo Ribeiro Jul 15 '14 at 16:50 — Rodrigo Ribeiro, Jul 15 '14 at 16:50
See this also, if you need details. – David Mitra Jul 15 '14 at 17:15 — David Mitra, Jul 15 '14 at 17:15

score 3 · Answer 1 · answered Jul 15 '14 at 16:58

3

Note that $$\left|\int_\Bbb R |f|-|f_n|\right|\leqslant \int_\Bbb R ||f|-|f_n||\leqslant \int_\Bbb R |f-f_n|=\lVert f-f_n\rVert$$

This shows one direction.

For the other direction, use Fatou's lemma with $g_n=|f_n|+|f|-|f-f_n|\geqslant 0$, $g=\liminf g_n$.

answered Jul 15 '14 at 16:58

Pedro

122,002

Credits to Scoccó. – Pedro Jul 15 '14 at 23:30

$f_n, f \in L^1$ and pointwise convergence implies ...

1 Answers1