Monotone Convergence Theorem for Positive Simple Functions

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

Let $f: X \to \overline \R \in \MM_{\overline \R}^+$ be a positive simple function.

Let $\EE^+$ be the space of positive simple functions.

For each $n \in \N$, let $f_n : X \to \R$ be a positive simple function, such that the sequence $\sequence {f_n}$ has:


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

and:


 * for each $x \in X$, the sequence $\sequence {\map {f_n} x}$ is increasing

where $\lim$ denotes a pointwise limit.

Then:


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

where the integral signs denote $\mu$-integration.

Proof
Note that since:


 * for each $x \in X$, the sequence $\sequence {\map {f_n} x}$ is increasing

we have that:


 * $f_i \le f_j$

whenever $i \le j$.

From Monotone Convergence Theorem (Real Analysis): Increasing Sequence, we further obtain:


 * $f_i \le f_j \le f$

whenever $i \le j$.

From Integral of Positive Measurable Function is Monotone, we have:


 * $\ds \int f_i \rd \mu \le \int f_j \rd \mu \le \int f \rd \mu$

So the sequence:


 * $\ds \sequence {\int f_n \rd \mu}$

is increasing and bounded.

So, by Monotone Convergence Theorem (Real Analysis): Increasing Sequence, it converges with:


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

Let $0 < \epsilon < 1$, we will construct a non-increasing sequence of positive simple functions $\sequence {g_n}$ such that:


 * $g_n \le f_n$

and:


 * $\ds \lim_{n \to \infty} \int g_n \rd \mu = \paren {1 - \epsilon} \int f \rd \mu$

Then from:


 * $\ds \int g_n \rd \mu \le \int f_n \rd \mu$

we will have:


 * $\ds \lim_{n \to \infty} \int g_n \rd \mu \le \lim_{n \to \infty} \int f_n \rd \mu$

giving:


 * $\ds \paren {1 - \epsilon} \int f \rd \mu \le \lim_{n \to \infty} \int f_n \rd \mu$

Since $\epsilon$ is arbitrary, we will then have:


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

giving the result.

From Simple Function has Standard Representation:


 * there exists disjoint $\Sigma$-measurable sets $E_1, E_2, \ldots, E_n$ and non-negative real numbers $a_1, a_2, \ldots, a_n$ such that:


 * $\ds \map f x = \sum_{i \mathop = 1}^n a_i \map {\chi_{E_i} } x$

For each $n \in \N$ and $i \in \N$, define:


 * $A_{n, i} = \set {x \in A_i : \map {f_n} x \ge \paren {1 - \epsilon} a_i}$

Since $f_n \le f_{n + 1}$, we have:


 * $\set {x \in A_i : \map {f_n} x \ge \paren {1 - \epsilon} a_i} \subseteq \set {x \in A_i : \map {f_{n + 1} } x \ge \paren {1 - \epsilon} a_i}$

We also have:


 * $\ds A_i = \bigcup_{n \mathop = 1}^\infty A_{n, i}$