Monotone class theorem

In measure theory and probability, the monotone class theorem connects monotone classes and sigma-algebras. The theorem says that the smallest monotone class containing an algebra of sets G is precisely the smallest σ-algebra containing G. It is used as a type of transfinite induction to prove many other theorems, such as Fubini's theorem.

Definition of a monotone class

A monotone class in a set $R$ is a collection $M$ of subsets of $R$ which contains $R$ and is closed under countable monotone unions and intersections, i.e. if $A_i \in M$ and $A_1 \subset A_2 \subset \ldots$ then $\cup_{i = 1}^\infty A_i \in M$, and similarly for intersections of decreasing sequences of sets.

Monotone class theorem for sets

Statement

Let G be an algebra of sets and define M(G) to be the smallest monotone class containing G. Then M(G) is precisely the σ-algebra generated by G, i.e. σ(G) = M(G)

Proof

The following was taken from Probability Essentials, by Jean Jacod and Philip Protter.[1] The idea is as follows: we know that the sigma-algebra generated by an algebra of sets G contains the smallest monotone class generated by G. So, we seek to show that the monotone class generated by G is in fact a sigma-algebra, which would then show the two are equal.

To do this, we first construct monotone classes that correspond to elements of G, and show that each equals the M(G), the monotone class generated by G. Using this, we show that the monotone classes corresponding to the other elements of M(G) are also equal to M(G). Finally, we show this result implies M(G) is indeed a sigma-algebra.

Let $\mathcal{B} = M(G)$, i.e. $\mathcal{B}$ is the smallest monotone class containing $G$. For each set $B$, denote $\mathcal{B}_B$ to be the collection of sets $A \in \mathcal{B}$ such that $A \cap B \in \mathcal{B}$. It is plain to see that $\mathcal{B}_B$ is closed under increasing limits and differences.

Consider $B \in G$. For each $C \in G$, $B \cap C \in G \subset \mathcal{B}$, hence $C \in \mathcal{B}_B$ so $G \subset \mathcal{B}_B$. This yields $\mathcal{B}_B = \mathcal{B}$ when $B \in G$, since $\mathcal{B}_B$ is a monotone class containing $G$, $\mathcal{B}_B \subset \mathcal{B}$ and $\mathcal{B}$ is the smallest monotone class containing $G$

Now, more generally, suppose $B \in \mathcal{B}$. For each $C \in G$, we have $B \in \mathcal{B}_C$ and by the last result, $B \cap C \in \mathcal{B}$. Hence, $C \in \mathcal{B}_B$ so $G \subset \mathcal{B}_B$, and so $\mathcal{B}_B = \mathcal{B}$ for all $B \in \mathcal{B}$ by the argument in the paragraph directly above.

Since $\mathcal{B}_B = \mathcal{B}$ for all $B \in \mathcal{B}$, it must be that $\mathcal{B}$ is closed under finite intersections. Furthermore, $\mathcal{B}$ is closed by differences, so it is also closed under complements. Since $\mathcal{B}$ is closed under increasing limits as well, it is a sigma-algebra. Since every sigma-algebra is a monotone class, $\mathcal{B} = \sigma\,(G)$, i.e. $\mathcal{B}$ is the smallest sigma-algebra containing G

Monotone class theorem for functions

Statement

Let $\mathcal{A}$ be a π-system that contains $\Omega\,$ and let $\mathcal{H}$ be a collection of functions from $\Omega$ to R with the following properties:

(1) If $A \in \mathcal{A}$, then $\mathbf{1}_{A} \in \mathcal{H}$

(2) If $f,g \in \mathcal{H}$, then $f+g$ and $cf \in \mathcal{H}$ for any real number $c$

(3) If $f_n \in \mathcal{H}$ is a sequence of non-negative functions that increase to a bounded function $f$, then $f \in \mathcal{H}$

Then $\mathcal{H}$ contains all bounded functions that are measurable with respect to $\sigma(\mathcal{A})$, the sigma-algebra generated by $\mathcal{A}$

Proof

The following argument originates in Rick Durrett's Probability: Theory and Examples. [2]

The assumption $\Omega\, \in \mathcal{A}$, (2) and (3) imply that $\mathcal{G} = \{A: \mathbf{1}_{A} \in \mathcal{H}\}$ is a λ-system. By (1) and the π − λ theorem, $\sigma(\mathcal{A}) \subset \mathcal{G}$. (2) implies $\mathcal{H}$ contains all simple functions, and then (3) implies that $\mathcal{H}$ contains all bounded functions measurable with respect to $\sigma(\mathcal{A})$

Results and Applications

As a corollary, if G is a ring of sets, then the smallest monotone class containing it coincides with the sigma-ring of G.

By invoking this theorem, one can use monotone classes to help verify that a certain collection of subsets is a sigma-algebra.

The monotone class theorem for functions can be a powerful tool that allows statements about particularly simple classes of functions to be generalized to arbitrary bounded and measurable functions.

References

1. ^ Jacod, Jean; Protter, Phillip (2004). Probability Essentials. Springer. p. 36. ISBN 978-3-540-438717.
2. ^ Durrett, Rick (2010). Probability: Theory and Examples (4th ed.). Cambridge University Press. p. 100. ISBN 978-0521765398.