Measurable Function is Pointwise Limit of Simple Functions

== Theorem ==
Let $\struct {X, \Sigma}$ be a measurable space.
Let $f: X \to \overline \R$ be a $\Sigma$-measurable function.
Then there exists a sequence $\sequence {f_n}_{n \mathop \in \N} \in \map \EE \Sigma$ of simple functions, such that:
:$\forall x \in X: \map f x = \ds \lim_{n \mathop \to \infty} \map {f_n} x$
That is, such that $f = \ds \lim_{n \mathop \to \infty} f_n$ pointwise.
If $f \ge 0$, the sequence $\sequence {f_n}_{n \mathop \in \N}$ may furthermore be taken to be increasing.
== Proof ==
First, let us prove the theorem when $f$ is a positive $\Sigma$-measurable function.
Now for any $n \in \N$, define for $0 \le k \le n 2^n$:
:${A_k}^n := \begin{cases}
 \set {k 2^{-n} \le f < \paren {k + 1} 2^{-n} } & : k \ne n 2^n \\
 \set {f \ge n} & : k = n 2^n
\end{cases}$
where for example $\set {f \ge n}$ is short for $\set {x \in X: \map f x \ge n}$.
It is immediate that the ${A_k}^n$ are pairwise disjoint, and that:
:$\ds \bigcup_{k \mathop = 0}^{n 2^n} {A_k}^n = X$
Subsequently, define $f_n: X \to \overline \R$ by:
:$\map {f_n} x := \ds \sum_{k \mathop = 0}^{n 2^n} k 2^{-n} \map {\chi_{ {A_k}^n} } x$
where $\chi_{ {A_k}^n}$ is the characteristic function of ${A_k}^n$.
Now if $\map f x < n$, then we have for some $k < n 2^{-n}$:
:$x \in {A_k}^n$
so that:
since:
:$x \in {A_k}^n$ $k 2^{-n} \le \map f x < \paren {k + 1} 2^{-n}$.
In particular, since $\map {f_n} x \le n$ for all $x \in X$, we conclude that pointwise, $f_n \le f$, for all $n \in \N$.
By Characterization of Measurable Functions and Sigma-Algebra Closed under Intersection, it follows that:
:${A_{n 2^n} }^n = \set {f \ge n}$
:${A_k}^n = \set {f \ge k 2^{-n} } \cap \set {f < \paren {k + 1} 2^{-n} }$
are all $\Sigma$-measurable sets.
Hence, by definition, all $f_n$ are $\Sigma$-simple functions.
It remains to show that:
:$\ds \lim_{n \mathop \to \infty} f_n = f$ pointwise.
Let $x \in X$ be arbitrary.
If $\map f x = +\infty$, then for all $n \in \N$, $x \in {A_{n 2^n} }^n$, so that:
:$\map {f_n} x = n$
Now clearly:
:$\ds \lim_{n \mathop \to \infty} n = +\infty$
showing convergence for these $x$.
If:
:$\map f x < +\infty$
then for some $n \in \N$:
:$\map f x < n$
By the reasoning above, we then have for all $m \ge n$:
:$\size {\map f x - \map {f_m} x} < 2^{-m}$
which by Sequence of Powers of Number less than One implies that:
:$\ds \lim_{n \mathop \to \infty} \map {f_n} x = \map f x$
Thus $\ds \lim_{n \mathop \to \infty} f_n = f$ pointwise.
This establishes the result for positive measurable $f$.
For arbitrary $f$, by Difference of Positive and Negative Parts, we have:
:$f = f^+ - f^-$
where $f^+$ and $f^-$ are the positive and negative parts of $f$.
By Function Measurable iff Positive and Negative Parts Measurable, $f^+$ and $f^-$ are positive measurable functions.
Thus we find sequences ${f_n}^+$ and ${f_n}^-$ converging pointwise to $f^+$ and $f^-$, respectively.
The Sum Rule for Real Sequences implies that for all $x \in X$:
:$\ds \lim_{n \mathop \to \infty} \map { {f_n}^+} x - \map { {f_n}^-} x = \map {f^+} x - \map {f^-} x = \map f x$
Furthermore, we have for all $n \in \N$ and $x \in X$:
Hence the result.
== Also see ==
* Bounded Measurable Function is Uniform Limit of Simple Functions, a strengthening when $f$ is bounded
== Sources ==
* : $8.8$
Category:Measurable Functions
Category:Simple Functions