- Let Q be the unit square in \(\mathbb { R } ^ { 2 }\) . Consider functions \(f _ { n } \in L ^ { 1 } ( Q )\) such that
\(f _ { n } \to f\) almost everywhere in Q and \(\int _ { Q } | f _ { n } | \to \int _ { Q } | f | < \infty .\)
Prove that \(\textstyle \int _ { A } | f _ { n } | \to \int _ { A } | f |\) for every measurable subset A of \(Q\)
Prove that \(f _ { n } \to f\) in \(L ^ { 1 }\)
Solution
1. (a) For measurable \(A\subseteq Q\), \(\int_A\abs{f_n}\to\int_A\abs f\).
1.1. \(\int_A\abs f\le\liminf_n\int_A\abs{f_n}\) and \(\int_{Q\setminus A}\abs f\le\liminf_n\int_{Q\setminus A}\abs{f_n}\).
Fatou’s lemma applied to \(\abs{f_n}\mathbf 1_A\) and \(\abs{f_n}\mathbf 1_{Q\setminus A}\), which converge a.e. to \(\abs f\mathbf 1_A\) and \(\abs f\mathbf 1_{Q\setminus A}\).
1.2. \(\limsup_n\int_A\abs{f_n}\le\int_A\abs f\).
Since \(\int_Q\abs{f_n}\to\int_Q\abs f\), step 1 gives \(\limsup_n\int_A\abs{f_n}=\int_Q\abs f-\liminf_n\int_{Q\setminus A}\abs{f_n}\le\int_Q\abs f-\int_{Q\setminus A}\abs f=\int_A\abs f\).
1.3. Q.E.D.
Steps 1 and 2.
2. (b) \(\int_Q\abs{f_n-f}\to0\).
The functions \(g_n\da\abs{f_n}+\abs f-\abs{f_n-f}\) are nonnegative and converge a.e. to \(2\abs f\). Fatou’s lemma and \(\int_Q\abs{f_n}\to\int_Q\abs f\) give \[2\int_Q\abs f\le\liminf_n\int_Qg_n=2\int_Q\abs f-\limsup_n\int_Q\abs{f_n-f}.\] Since \(\int_Q\abs f<\infty\), \(\limsup_n\int_Q\abs{f_n-f}\le0\).
3. Q.E.D.
Steps 1 and 2 prove parts (a) and (b).