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From Donald Cohn's Measure Theory, section 2.4, exercise 10.

Let $(X, A, \mu)$ be a measure space, and let $f$ and $f_1, f_2, \dots$ be non-negative functions that belong to $L^1(X, A, \mu, R)$ and satisfy

(i) $\{f_n\}_n$ converges to $f$ almost everywhere;

(ii) $\int fd\mu = \lim_n\int f_nd\mu$.

Show that $\lim_n\int |f_n - f|d\mu = 0$.

I let $f_n = 1/n$ on $[n, n+1]$ and $0$ elsewhere, and let $f=0$, then all the convergence theorems in that section (dominated convergence, and monotone convergence) failed.

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  • $\begingroup$ What have you tried? There basically only 3 convergence theorems for Lebesgue integral. $\endgroup$ – user251257 Jan 23 '17 at 23:52
  • $\begingroup$ I let f_n = 1/n on [n, n+1] and 0 elsewhere, then all the convergence theorems in that section (dominated convergence, and monotone convergence) failed. $\endgroup$ – MaudPieTheRocktorate Jan 23 '17 at 23:55
  • $\begingroup$ They converge to f = 0. |f_n - f| integrates to 1/n which converges to 0. $\endgroup$ – MaudPieTheRocktorate Jan 24 '17 at 0:09
  • $\begingroup$ What's the third convergence theorem you mentioned? $\endgroup$ – MaudPieTheRocktorate Jan 24 '17 at 0:11
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Hint:

Fatou's Lemma yields $$ 2\int f = \int \liminf_{n\to\infty} \left ( f_n + f - |f_n - f| \right ) \le \int f + \int f + \liminf_{n\to\infty} \left ( - \int |f_n - f| \right ). $$

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Here is a solution using the dominated convergence theorem:

Let $F_n = f_n-|f_n-f|$. Then $F_n$ converges pointwise to $f$ and we have by the reverse triangle inequality $|F_n| = |f_n-|f_n-f|| = ||f_n|-|f_n-f||\leq |f|$. Hence by the dominated convergence theorem $\lim_n \int F_n = \int f$. But then $$\lim_n \int |f_n-f| = \lim_n \int (|f_n-f|-f_n+f_n) = -\lim_n \int F_n + \lim_n\int f_n = 0$$ since $\lim_n \int f_n = \int f$.

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