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How do I prove the following theorem of dominated convergence type?

If $g_n,f_n,f,g \in L^1$, $f_n \to f$ and $g_n \to g$ a.e., $|f_n|\le g_n$ and $\int g_n \to \int g$, then $\int f_n \to \int f$.

Can we not assume $f,g \in L^1$ and still have this result?

Gono
  • 5,598

1 Answers1

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From your assumptions you have $0 \le g_n + f_n$ and $0 \le g_n - f_n$ so we get with Fatou's Lemma

$$\int (g + f) d\mu \le \liminf_{n\to\infty} \int (g_n + f_n) d\mu = \int g d\mu + \liminf_{n\to\infty} \int f_n d\mu \quad(*)$$

and also:

$$\int (g - f) d\mu \le \liminf_{n\to\infty} \int (g_n - f_n) d\mu = \int g d\mu - \limsup_{n\to\infty} \int f_n d\mu \quad(**)$$

Substract $\int g d\mu$ on every side leads us to: $$\int f d\mu \le \liminf_{n\to\infty} \int f_n d\mu \le \limsup_{n\to\infty} \int f_n d\mu \le \int f d\mu$$

So we're done…

Gono
  • 5,598