I have the following problem:

Let $(\Omega,F,\mu)$ be a measure space and $(f_n)_{n\in\mathbb{N}}$ be a sequence of non-negative integrable functions, so that $\lim_{n\rightarrow \infty} \int f_n d\mu$ exists. We define $f:=\lim f_n$, which exists. Show f is integrable and $$\lim_{n\rightarrow\infty} \int |f_n-f|d\mu=\lim_{n\rightarrow\infty}\int f_n d\mu-\int f d\mu$$

This reminds me of the dominated convergence theorem. But this theorem needs an integrable function $g$ with $|f(x)|\leq g(x)$. Why don't we need this here? Is this even connected to this theorem? Thanks in advance.

  • $\begingroup$ Thank you, I just edited it. $\endgroup$ – Tobi92sr Jun 5 '18 at 23:52
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    $\begingroup$ If n is large enough, $2f_n\gt f$ implies f is integrable? $\endgroup$ – herb steinberg Jun 5 '18 at 23:57
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    $\begingroup$ @herbsteinberg : I don't think it is possible to conclude $2f_n > f$ for all sufficiently large $n$. On the other hand, Fatou's lemma for nonnegative functions seems to immediately imply that $\int f <\infty$. (by the way I assume "the limit exists" means "the limit exists and is finite"). $\endgroup$ – Michael Jun 6 '18 at 2:56
  • $\begingroup$ Convergence theorems tell us when $\int f_n \rightarrow \int f$, that is, they tell us when the right-hand-side of the desired equality is zero. This result is different and holds even when there are no conditions available to ensure the right-hand-side is zero. $\endgroup$ – Michael Jun 6 '18 at 4:13

You can first prove that $\int f < \infty$ by using Fatou's lemma on the nonnegative functions $f_n$.

You can then prove the desired inequality in two stages:

1) Prove $\liminf_{n\rightarrow\infty} \int |f_n-f| \geq \lim_{n\rightarrow\infty} \int f_n - \int f$

2) Define $g_n(x) = \inf_{m\geq n} f_m(x)$, note that $g_n(x)\nearrow f(x)$, and use the monotone convergence theorem to prove the remaining inequality $$\limsup_{n\rightarrow\infty}\int |f_n-f| \leq \lim_{n\rightarrow\infty} \int f_n - \int f$$

  • $\begingroup$ Thanks for the hints. I showed: $f$ is integrable and $1)$. But I don't see how to show $2)$ using the monotone convergence theorem. Could you elaborate? $\endgroup$ – Tobi92sr Jun 6 '18 at 18:51
  • $\begingroup$ You can exploit triangle and other inequality relationships between $f_n, f, g_n$. $\endgroup$ – Michael Jun 6 '18 at 22:15

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