Monotone Convergence Theorem (Measure Theory)

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This proof is about convergence of mappings in measure theory. For other uses, see Monotone Convergence Theorem.

Theorem

Let $\left({X, \Sigma, \mu}\right)$ be a measure space.

Let $\left({u_n}\right)_{n \in \N} \in \mathcal L^1 \left({\mu}\right)$, $u_n: X \to \R$ 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:

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

and, in that case:

$\displaystyle \int \sup_{n \mathop \in \N} u_n \, \mathrm d \mu = \sup_{n \mathop \in \N} \int u_n \, \mathrm d \mu$


Corollary

Let $\left({u_n}\right)_{n \in \N} \in \mathcal{L}^1 \left({\mu}\right)$, $u_n: X \to \R$ 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:

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

and, in that case:

$\displaystyle \int \inf_{n \mathop \in \N} u_n \, \mathrm d \mu = \inf_{n \mathop \in \N} \int u_n \, \mathrm d \mu$


Proof


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