Exercise 1.5.17 (Defect version of Fatou's lemma)

Question

Let $\left(fight)_{n\in\mathbb{N}}:X	o [0,+\infty)$ be a sequence of measurable unsigned functions such that $\sup\int_X f_n \,\mathrm{d}\mu <\infty$, and converge pointwise almost everywhere to some measurable limit $f:X	o [0,+\infty)$. Show that $f_n$ converges in $L^1$ norm if and only if $\int_X f_n \mathrm{d}\mu$ converges to $\int_{X} f \,\mathrm{d}{\mu}$.

Elu Thingol · Accepted Answer

$
ewcommand{\intx}[1]{\int_{X} #1 \mathop{}\!\mathrm{d}{\mu}}$
\begin{itemize}
\item[$\implies)$] By Fatou's lemma (Corollary 1.4.46), we have      $$0 \leq \lim_{n	o\infty} \intx{f_n} - \int_{X} f \,\mathrm{d}{\mu} = \lim_{n	o\infty} \intx{(f_n-f)} \leq \lim_{n	o\infty} \intx{|f_n-f|} = \lim_{n	o\infty} \|f_n-f\| = 0.$$

\item[$\impliedby)$] Since $\lim_{n	o\infty}f_n= f$ pointwise, it is also obvious that $\lim_{n	o\infty}|f_n-f| = 0$ pointwise. We thus have, by Fatou's lemma
$$\intx{2f} = \intx{\lim_{n	o\infty} (f_n+f-|f_n-f|)} \stackrel{FL}{\leq} \lim_{n	o\infty} \intx{\left(f_n+f-|f_n-f|ight)} = \int_{X} f \,\mathrm{d}{\mu} + \lim_{n	o\infty}\intx{f_n} - \lim_{n	o\infty}\intx{|f_n-f|}.$$
Cancelling $\int 2f$ out, we obtain the desired result.
\end{itemize}

Exercise 1.5.17 (Defect version of Fatou's lemma)

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Exercise 1.5.17 (Defect version of Fatou's lemma)

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