Discussion
Let \((\Omega, \mathbf{F}, \mu)\) be a measure space and let \(f_{n} : \Omega \to \bar{\mathbf{R}}\), be a sequence of extended Borel measurable functions. Suppose \(|f_{n}| \le 100\), \(\forall\ n = 1, 2, ......,\) and \(\lim_{n\to\infty} f_{n} = f\) a.e. \(\mu\). Then, which of the following statements is false ?
Answer: B
The false statement is \(\lim \int |f_{n}|\,d\mu = \int |f|\,d\mu\).
The uniform bound 100 is not an integrable dominating function unless \(\mu(\Omega) \lt \infty\), and the measure space here is arbitrary. On \(\mathbf{R}\) with Lebesgue measure, \(f_{n} = 1_{[n, n+1]}\) satisfies \(|f_{n}| \le 1\) and \(f_{n} \to 0\) everywhere, yet \(\int |f_{n}| = 1\) for every \(n\), while \(\int |f| = 0\).
The others always hold: A is Fatou's lemma; C holds because \(\limsup|f_{n}| = |f|\) almost everywhere; and D is the monotone convergence theorem applied to the increasing partial sums of \(|f_{k}|\).
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