Prove Fatou’s lemma using the Monotone Convergence Theorem.
Show that if \(\theset{f_n}\) is in \(L^1\) and \(\sum \int \abs{f_n} < \infty\) then \(\sum f_n\) converges to an \(L^1\) function and \[\int \sum f_n = \sum \int f_n.\]
Solution
1. For measurable \(f_n \ge 0\), \(\int \liminf_n f_n \le \liminf_n \int f_n\).
1.1. \(g_k = \inf_{n \ge k} f_n\) is measurable, and \(g_k \uparrow \liminf_n f_n\) pointwise.
\(g_k\) is an infimum of countably many measurable functions, \(g_k \le g_{k+1}\), and \(\sup_k g_k = \liminf_n f_n\) by definition.
1.2. \(\int g_k \le \inf_{n \ge k}\int f_n\).
\(g_k \le f_n\) for \(n \ge k\), and the integral is monotone.
1.3. Q.E.D.
By the monotone convergence theorem and steps 1 and 2, \(\int \liminf_n f_n = \lim_k \int g_k \le \lim_k \inf_{n \ge k}\int f_n = \liminf_n \int f_n\).
2. If \(f_n \in L^1\) and \(\sum_n \int |f_n| < \infty\), then \(\sum_n f_n\) converges a.e. to an \(L^1\) function \(F\) and \(\int F = \sum_n \int f_n\).
2.1. \(g = \sum_n |f_n|\) satisfies \(\int g = \sum_n \int |f_n| < \infty\), so \(g < \infty\) a.e.
The first equality is the monotone convergence theorem for the partial sums. A nonnegative function with finite integral is finite a.e.
2.2. Q.E.D.
Where \(g(x) < \infty\) the series \(\sum_n f_n(x)\) converges absolutely; let \(F(x)\) be its sum there and \(F = 0\) elsewhere. Then \(|F| \le g\), so \(F \in L^1\). For every \(N\), \[ \left|\int F - \sum_{n=1}^N \int f_n\right| \le \int \Big|\sum_{n>N} f_n\Big| \le \int \sum_{n>N}|f_n| = \sum_{n>N}\int|f_n|, \] using the monotone convergence theorem in the last step. The right side tends to \(0\).