Monotone Convergence Theorem
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Theorem
Monotone Convergence Theorem (Real Analysis)
Let $\sequence {x_n}$ be a bounded monotone sequence sequence in $\R$.
Then $\sequence {x_n}$ is convergent.
Monotone Convergence Theorem (Measure Theory)
Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $u : X \to \overline \R_{\ge 0}$ be a positive $\Sigma$-measurable function.
Let $\sequence {u_n}_{n \mathop \in \N}$ be an sequence of positive $\Sigma$-measurable functions $u_n : X \to \overline \R_{\ge 0}$ such that:
- $\map {u_i} x \le \map {u_j} x$ for all $i \le j$
and:
- $\ds \map u x = \lim_{n \mathop \to \infty} \map {u_n} x$
hold for $\mu$-almost all $x \in X$.
Then:
- $\ds \int u \rd \mu = \lim_{n \mathop \to \infty} \int u_n \rd \mu$