# proof of functional monotone class theorem

We start by proving the following version of the monotone class theorem.

###### Theorem 1.

Let $(X,\mathcal{A})$ be a measurable space and $\mathcal{S}$ be a $\pi$-system (http://planetmath.org/PiSystem) generating the $\sigma$-algebra (http://planetmath.org/SigmaAlgebra) $\mathcal{A}$. Suppose that $\mathcal{H}$ be a vector space of real-valued functions on $X$ containing the constant functions and satisfying the following,

• if $f\colon X\rightarrow\mathbb{R}_{+}$ is bounded and there is a sequence of nonnegative functions $f_{n}\in\mathcal{H}$$f$, then $f\in\mathcal{H}$.

• for every set $A\in\mathcal{S}$$1_{A}$ is in $\mathcal{H}$.

Then, $\mathcal{H}$ contains every bounded and measurable function from $X$ to $\mathbb{R}$.

Let $\mathcal{D}$ consist of the collection of subsets $B$ of $X$ such that the characteristic function $1_{B}$ is in $\mathcal{H}$. Then, by the conditions of the theorem, the constant function $1_{X}$ is in $V$ so that $X\in\mathcal{D}$, and $\mathcal{S}\subseteq\mathcal{D}$. For any $A\subseteq B$ in $\mathcal{D}$ then $1_{B\setminus A}=1_{B}-1_{A}\in\mathcal{H}$, as $\mathcal{H}$ is closed under linear combinations, and therefore $B\setminus A$ is in $\mathcal{D}$. If $A_{n}\in\mathcal{D}$ is an increasing sequence, then $1_{A_{n}}\in\mathcal{H}$ increases pointwise to $1_{\bigcup_{n}A_{n}}$, which is therefore in $\mathcal{H}$, and $\bigcup_{n}A_{n}\in\mathcal{D}$. It follows that $\mathcal{D}$ is a Dynkin system, and Dynkin’s lemma shows that it contains the $\sigma$-algebra $\mathcal{A}$.

We have shown that $1_{A}\in\mathcal{H}$ for every $A\in\mathcal{A}$. Now consider any bounded and measurable function $f\colon X\rightarrow\mathbb{R}$ taking values in a finite set $S\subseteq\mathbb{R}$. Then,

 $f=\sum_{s\in S}s1_{f^{-1}(\{s\})}$

is in $\mathcal{H}$.

We denote the floor function by $\lfloor\cdot\rfloor$. That is, $\lfloor a\rfloor$ is defined to be the largest integer less than or equal to the real number $a$. Then, for any nonnegative bounded and measurable $f\colon X\rightarrow\mathbb{R}$, the sequence of functions $f_{n}(x)=2^{-n}\lfloor 2^{n}f(x)\rfloor$ each take values in a finite set, so are in $\mathcal{H}$, and increase pointwise to $f$. So, $f\in\mathcal{H}$.

Finally, as every measurable and bounded function $f\colon X\rightarrow\mathbb{R}$ can be written as the difference of its positive and negative parts $f=f_{+}-f_{-}$, then $f\in\mathcal{H}$.

We now extend this result to prove the following more general form of the theorem.

###### Theorem 2.

Let $X$ be a set and $\mathcal{K}$ be a collection of bounded and real valued functions on $X$ which is closed under multiplication, so that $fg\in\mathcal{K}$ for all $f,g\in\mathcal{K}$. Let $\mathcal{A}$ be the $\sigma$-algebra on $X$ generated by $\mathcal{K}$.

Suppose that $\mathcal{H}$ is a vector space of bounded real valued functions on $X$ containing $\mathcal{K}$ and the constant functions, and satisfying the following

• if $f\colon X\rightarrow\mathbb{R}$ is bounded and there is a sequence of nonnegative functions $f_{n}\in\mathcal{H}$ increasing pointwise to $f$, then $f\in\mathcal{H}$.

Then, $\mathcal{H}$ contains every bounded and real valued $\mathcal{A}$-measurable function on $X$.

Let us start by showing that $\mathcal{H}$ is closed under uniform convergence. That is, if $f_{n}$ is a sequence in $\mathcal{H}$ and $\|f_{n}-f\|\equiv\sup_{x}|f_{n}(x)-f(x)|$ converges to zero, then $f\in\mathcal{H}$. By passing to a subsequence if necessary, we may assume that $\|f_{n}-f_{m}\|\leq 2^{-n}$ for all $m\geq n$. Define $g_{n}\equiv f_{n}-2^{1-n}+2+\|f\|$. Then $g_{n}\in\mathcal{H}$ since $\mathcal{H}$ is a vector space containing the constant functions. Also, $g_{n}$ are nonnegative functions increasing pointwise to $f+2+\|f\|$ which must therefore be in $\mathcal{H}$, showing that $f\in\mathcal{H}$ as required.

Now let $\mathcal{H}_{0}$ consist of linear combinations of constant functions and functions in $\mathcal{K}$ and $\mathcal{\bar{H}}_{0}$ be its closure (http://planetmath.org/Closure) under uniform convergence. Then $\mathcal{\bar{H}}_{0}\subseteq\mathcal{H}$ since we have just shown that $\mathcal{H}$ is closed under uniform convergence. As $\mathcal{K}$ is already closed under products, $\mathcal{H}_{0}$ and $\mathcal{\bar{H}}_{0}$ will also be closed under products, so are algebras (http://planetmath.org/Algebra). In particular, $p(f)\in\mathcal{\bar{H}}_{0}$ for every $f\in\mathcal{\bar{H}}_{0}$ and polynomial $p\in\mathbb{R}[X]$. Then, for any continuous function $p\colon\mathbb{R}\rightarrow\mathbb{R}$, the Weierstrass approximation theorem says that there is a sequence of polynomials $p_{n}$ converging uniformly to $p$ on bounded intervals, so $p_{n}(f)\rightarrow p(f)$ uniformly. It follows that $p(f)\in\mathcal{\bar{H}}_{0}$. In particular, the minimum of any two functions $f,g\in\mathcal{\bar{H}}_{0}$, $f\wedge g=f-|f-g|$ and the maximum $f\vee g=f+|g-f|$ will be in $\mathcal{\bar{H}}_{0}$.

We let $\mathcal{S}$ consist of the sets $A\subseteq X$ such that there is a sequence of nonnegative $f_{n}\in\mathcal{\bar{H}}_{0}$ increasing pointwise to $1_{A}$. Once it is shown that this is a $\pi$-system generating the $\sigma$-algebra $\mathcal{A}$, then the result will follow from theorem 1.

If $f_{n},g_{n}\in\mathcal{\bar{H}}_{0}$ are nonnegative functions increasing pointwise to $1_{A},1_{B}$ then $f_{n}g_{n}$ increases pointwise to $1_{A\cap B}$, so $A\cap B\in\mathcal{S}$ and $\mathcal{S}$ is a $\pi$-system.

Finally, choose any $f\in\mathcal{K}$ and $a\in\mathbb{R}$. Then, $f_{n}=((n(f-a))\vee 0)\wedge 1$ is a sequence of functions in $\mathcal{\bar{H}}_{0}$ increasing pointwise to $1_{f^{-1}((a,\infty))}$. So, $f^{-1}((a,\infty))\in\mathcal{S}$. As intervals of the form $(a,\infty)$ generate the Borel $\sigma$-algebra on $\mathbb{R}$, it follows that $\mathcal{S}$ generates the $\sigma$-algebra $\mathcal{A}$, as required.

Title proof of functional monotone class theorem ProofOfFunctionalMonotoneClassTheorem 2013-03-22 18:38:44 2013-03-22 18:38:44 gel (22282) gel (22282) 4 gel (22282) Proof msc 28A20