Pratt's Lemma

Theorem
Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let:
 * $\sequence {g_n}_{n \mathop \in \N}$


 * $\sequence {G_n}_{n \mathop \in \N}$


 * $\sequence {f_n}_{n \mathop \in \N}$

be sequences of $\mu$-integrable functions.

Let the pointwise limits:
 * $\ds f := \lim_{n \mathop \to \infty} f_n$


 * $\ds g := \lim_{n \mathop \to \infty} g_n$


 * $\ds G := \lim_{n \mathop \to \infty} G_n$

exist.

Let $g$ and $G$ be $\mu$-integrable.

Suppose that, for all $x \in X$ and $n \in \N$:


 * $\map {g_n} x \le \map {f_n} x \le \map {G_n} x$

Finally, suppose the following hold:


 * $\ds \lim_{n \mathop \to \infty} \int g_n \rd \mu = \int g \rd \mu$


 * $\ds \lim_{n \mathop \to \infty} \int G_n \rd \mu = \int G \rd \mu$

Then:


 * $\ds \lim_{n \mathop \to \infty} \int f_n \rd \mu = \int f \rd \mu$

and the latter is finite.