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I am working on a problem$^{(*)}$ on Lebesgue integral looks like this:

Given that both $f_n$ and $f$ are integrable, $f_n \longrightarrow f$ a.e., and $\int|f_n| \longrightarrow \int |f|$. Show that $$\int |f_n - f| \longrightarrow 0.$$

To me, the question "makes sense" since $\int |f_n - f| \longrightarrow 0$ implies $\lim_{n \to \infty} \int |f_n - f| = 0,$ and $|f_n - f|$ is approaching zero as $n$ is approaching infinity, therefore the integral is approaching zero. But I do not know how to say it mathematically.

Any help or hints would be very much appreciated. Thanks for your time.

(*) Richard F. Bass' Real Analysis, 2nd. edition, chapter 7: Limit Theorems, exercise 7.5, page 57.

A.Magnus
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  • This is a duplicate of http://math.stackexchange.com/questions/51502/convergence-a-e-and-of-norms-implies-that-in-lebesgue-space and also an easy consequence of the following more general statement: http://math.stackexchange.com/questions/72174/general-lebesgue-dominated-convergence-theorem. – PhoemueX Mar 28 '15 at 17:38

1 Answers1

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Let $g_n = |f_n| + |f| - |f_n - f|$. Then $g_n \ge 0$ and $g_n \to 2|f|$ a.e., hence

$$\liminf_{n\to \infty} \int g_n \ge \int 2|f|$$

by Fatou's lemma. Since $\int |f_n| \to \int |f|$,

$$\liminf_{n\to \infty} \int g_n = \liminf_{n\to \infty} \int (|f_n| + |f|) - \limsup_{n\to \infty} \int |f_n - f| = \int 2|f| - \limsup_{n\to \infty} \int |f_n - f|.$$

Thus

$$\int 2|f| - \limsup_{n\to \infty} \int |f_n - f| \ge \int 2|f|.$$

Deduce from this inequality and the fact $\int 2|f| = 2\int |f| < \infty$ that

$$\int |f_n - f| \to 0.$$

kobe
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