Monotone classes
A very important notion is that of a monotone class. We will see that there are many things which can be deduced by using the monotone class lemma.
Let [math]E[/math] be some topological space and let [math]M\subset\mathcal{P}(E)[/math]. [math]M[/math] is called a monotone class if the following holds.
- [math]E\in M[/math].
- Let [math]A\in M[/math] and [math]B\in M[/math]. If [math]A\subset B[/math] [math]\Longrightarrow B\setminus A\in M[/math].
- Let [math](A_n)_{n\in\N}\in M[/math]. If [math]A_n\subset A_{n+1}[/math] [math]\Longrightarrow \bigcup_{n\in\mathbb{N}}A_n \in M[/math].
A [math]\sigma[/math]-Algebra is a monotone class[a]
As for [math]\sigma[/math]-Algebras, we notice that an arbitrary intersection of monotone classes is again a monotone class. Thus, if [math]\mathcal{C}\in \mathcal{P}(E)[/math], we can define the monotone class generated by [math]\mathcal{C}[/math] as
This is also by construction the smallest monotone class containing [math]\mathcal{C}[/math].
Let [math]E[/math] be a topological space. If [math]\mathcal{C}\subset\mathcal{P}(E)[/math] is stable under finite intersection, i.e. for [math]A\in \mathcal{C}[/math] and [math]B\in\mathcal{C}\Longrightarrow A\cap B\in \mathcal{C}[/math], then
It is obvious that, [math]M(\mathcal{C})\subset\sigma(\mathcal{C})[/math] since a [math]\sigma[/math]-Algebra is also a monotone class. Next we want to show that [math]M(\mathcal{C})[/math] is a [math]\sigma[/math]-Algebra to conclude that [math]\sigma(\mathcal{C})\subset M(\mathcal{C})[/math] and hence then [math]M(\mathcal{C})[/math] contains [math]\mathcal{C}[/math], i.e. [math]\sigma(\mathcal{C})[/math]. It is not difficult to see that a monotone class, which is stable under finite intersection, is a [math]\sigma[/math]-Algebra. Let us therefore show that [math]M(\mathcal{C})[/math] is stable under finite intersections. First, we fix [math]A\in\mathcal{C}[/math] and define
Then we get that [math]\mathcal{C}\in M_A[/math] since [math]\mathcal{C}[/math] is stable under finite intersections and obviously [math]E\in M_A[/math]. We can also note that If [math]B,B'\in M_A[/math] and [math]B\subset B'[/math], with
then [math]B'\setminus B\in M_A[/math]. Moreover, if [math](B_n)_{n\in\N}\in M_A[/math] and [math]A\cap\bigcup_n\underbrace{(A\cap B_n)}_{\in M(\mathcal{C})}[/math] we get the implication
since [math](A\cap B_n)[/math] is increasing. Finally we can conclude the above facts. That means if [math]M_A[/math] is a monotone class containing [math]\mathcal{C}[/math], then [math]M_A=M(\mathcal{C})[/math], which shows that for all [math]A\in\mathcal{C}[/math] and [math]B\in M(\mathcal{C})[/math] we get [math]A\cap B\in M(\mathcal{C})[/math]. We can now apply the same idea another time. Fix [math]B\in M(\mathcal{C})[/math] and define
Let [math](E,\A)[/math] be a measurable space and let [math]\mu,\nu[/math] be two measures on [math](E,\mathcal{A})[/math]. Moreover, assume that there exists a family of subsets [math]\mathcal{C}[/math], which is stable under finite intersections, such that [math]\sigma(\mathcal{C})=\mathcal{A}[/math] and [math]\mu(A)=\nu(A)[/math] for all [math]A\in\mathcal{C}[/math]. Then the following hold.
- If [math]\mu(E)=\nu(E) \lt \infty[/math], then we get [math]\mu=\nu[/math].
- If there exists an increasing family [math](E_n)_{n\in\N}[/math] with [math]E_n\in\mathcal{C}[/math] such that
[[math]] E=\bigcup_{n\in\N}E_n [[/math]]and [math]\mu(E_n)=\nu(E_n) \lt \infty[/math], then it follows that [math]\mu=\nu[/math].
Let us first define the set [math]G:=\{A\in\A\mid \mu(A)=\nu(A)\}[/math]. By assumption we get that [math]\mathcal{C}\subset G[/math]. Moreover, we note that [math]G[/math] is a monotone class. Note at first that [math]E\in G[/math] by assumption since [math]\mu(E)=\nu(E)[/math]. Now let [math]A,B\in G[/math] such that [math]A\subset B[/math] and since
Now since [math]\mu(E)=\nu(E)[/math], we get the same for the sequence elements and obtain therefore that [math]\mu_n=\nu_n[/math]. Moreover, for [math]A\in \A[/math], we have
There are several applications of this corollary. Let us emphasize a first one, by giving already a small introduction to the Lebesgue measure. Assume that [math]\lambda[/math] is a measure on the measurable space [math](\R,\B(\R))[/math] such that [math]\lambda((a,b))=b-a[/math] for [math]a \lt b[/math] and let [math]\mathcal{C}[/math] be the class of intervals [math]E_n=(-n,n)[/math] for [math]n\geq 1[/math]. With the corollary above, it follows that [math]\lambda[/math] is unique. We will call [math]\lambda[/math] the Lebesgue measure. A second application is that a finite measure on [math](\R,\B(\R))[/math] is uniquely characterized, for [math]a\in\R[/math], by the values
General references
Moshayedi, Nima (2020). "Lectures on Probability Theory". arXiv:2010.16280 [math.PR].