A criterion for checking if a set of measurable functions satisfy a certain property.

Question

I have recently become interested in analysis, and while checking an old set of notes of mine, I discovered this "theorem" stated without proof in them:

Let $X$ be a compact metric space equipped with the Borel $\sigma$-algbera. Let $C_+(X)$ be the set of all the non-negative real-valued continuous functions on $X$. Let $BM_+(X)$ be the set of all the non-negative bounded measurable functions on $X$.

Theorem. Let $S\subseteq BM_+(X)$ be such that
$1)$ Whenever $f_1, f_2, f_3, \ldots\in S$, we have $\sum_if_i\in S$.
$2)$ Whenever $f, g\in S$ with $f\geq g$, we have $f-g\in S$.
$3)$ $C_+(X)\subseteq S$.
Then $BM_+(X)\subseteq S$.

The problem with the above statement is that the first condition on $S$ cannot be satisfied. For if $f\in S$ is such that $f(x)>0$ for some $x$, then setting $f_i=f$ for each $i\geq 1$ implies by (1) that a non-bounded function is in $F$, contrary to the fact that $F\subseteq BM_+(X)$.

So the theorem is not true (or rather vacuously true) as stated.

What is the modification that one needs to make in order to make this statement true?

I am interested in this because in some situations it may be easier to see that continuous functions satisfy a certain property. A theorem of the above type will help us conclude that all bounded measurable functions also satisfy that property, provided some reasonable conditions are satisfied.

If you happen to know the correct statement, then can you also please provide a reference?

Answer 1

$\newcommand{\set}[1]{\{#1\}}$ $\newcommand{\mc}{\mathcal}$ $\newcommand{\R}{\mathbf R}$

The correct formulation and the proof is presented as Lemma 2 below.

Let $X$ be a set and $\mc D$ be a collection of subsets of $X$. We say that $\mc D$ is a Dynkin system (or $\lambda$-system) if

$\bullet$ $X\in \mc D$.
$\bullet$ If $A\in \mc D$ then $X\setminus A\in \mc D$.
$\bullet$ If $A_1, A_2, A_3, \ldots \mc D$ such that $A_i\cap A_j=\emptyset$ whenever $i\neq j$, then we have $\bigcup_{i=1}^\infty A_i\in \mc D$.

Also, we way that a non-empty collection of subsets $\mc P$ of $X$ is a $\pi$-system if $\mc P$ is closed under finite intersections.

Theorem 1. Dynkin's $\pi$-$\lambda$ Theorem. Let $\mc P$ be a $\pi$-system and $\mc D$ be a $\lambda$-system on $X$ such that $\mc P\subseteq \mc D$. Then $\sigma(\mc P)\subseteq \mc D$, where $\sigma(\mc P)$ denotes the $\sigma$-algebra generated by $\mc P$.

Let $M_{\geq 0}(X)$ denote the set of all the non-negative measurable functions on $X$ and $C_{\geq 0}(X)$ denotes the set of all the non-negative continuous function on $X$.

Lemma 2. Let $S\subseteq M_{\geq 0}(X)$ be such that
1) $C_{\geq 0}(X)\subseteq S$.
2) If $f, g\in F$ with $g\geq f$, then $g-f\in S$.
3) If $f_1, f_2, f_3 , \ldots\in F$, then we have $\sum_{i=1}^\infty \in S$.
Then $S=M_{\geq 0}(X)$.

Proof. It is enough to show that the indicator functions of Borel sets in $X$ are in $S$. Let $\mc C=\set{A\in \mc X:\ \chi_A\in S}$. It is easy to see that $\mc C$ is a $\lambda$-system.

We show that each open set is in $\mc C$. Let $U$ be an arbitrary open set in $X$. For each $n\geq 1$, define $$K_n=\set{x\in X:\ d(x, X\setminus U) \geq 1/n}$$ Then each $K_n$ is compact and is contained in $U$. By Urysohn's lemma, we can find a continuous function $g_n:X\to [0, 1]$ supported in $U$ and which takes the value $1$ on all of $K_n$. For each $k\geq 1$, define $$ h_k=\max\set{g_1 , \ldots, g_k} $$ Then note that

$\bullet$ Each $h_k$ vanishes outside of $U$.
$\bullet$ $h_k(x)\uparrow 1$ if $x\in U$.
$\bullet$ $h_1\leq h_2\leq h_3\leq \cdots $

Finally define $q_1=h_1$, and $q_i=h_i-h_{i-1}$ for each $i\geq 2$. Then $\sum_{i}q_i=\chi_U$. But since each $q_i$ is in $S$, we have that $\chi_U$ is also in $S$. So the collection of all open sets, which is a $\pi$-system, is contained in $\mc C$. But then by Theorem 1 we have that $\mc C$ contains the $\sigma$-algebra generated by the open sets, and we are done. $\blacksquare$