Monotone Convergence Theorem (Measure Theory)

This proof is about Monotone Convergence Theorem in the context of Measure Theory. For other uses, see Monotone Convergence Theorem.

Theorem

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

Let $\sequence {u_n}_{n \mathop \in \N} \in \map {\LL^1} \mu$, $u_n: X \to \R$ be a increasing sequence of $\mu$-integrable functions.

Let $\displaystyle \sup_{n \mathop \in \N} u_n: X \to \overline \R$ be the pointwise supremum of the $u_n$.

Then $\displaystyle \sup_{n \mathop \in \N} u_n$ is $\mu$-integrable if and only if:

$\displaystyle \sup_{n \mathop \in \N} \int u_n \rd \mu < +\infty$

and, in that case:

$\displaystyle \int \sup_{n \mathop \in \N} u_n \rd \mu = \sup_{n \mathop \in \N} \int u_n \rd \mu$

Corollary

Let $\sequence {u_n}_{n \mathop \in \N} \in \map {\LL^1} \mu$, $u_n: X \to \R$ be a decreasing sequence of $\mu$-integrable functions.

Let $\displaystyle \inf_{n \mathop \in \N} u_n: X \to \overline \R$ be the pointwise infimum of the $u_n$.

Then $\displaystyle \inf_{n \mathop \in \N} u_n$ is $\mu$-integrable if and only if:

$\displaystyle \inf_{n \mathop \in \N} \int u_n \rd \mu > -\infty$

and, in that case:

$\displaystyle \int \inf_{n \mathop \in \N} u_n \rd \mu = \inf_{n \mathop \in \N} \int u_n \rd \mu$