Monotone class

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In measure theory and probability, the monotone class theorem connects monotone classes and 𝜎-algebras. The theorem says that the smallest monotone class containing an algebra of sets G {\displaystyle G} {\displaystyle G} is precisely the smallest 𝜎-algebra containing  G . {\displaystyle G.} {\displaystyle 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 is a family (i.e. class) M {\displaystyle M} {\displaystyle M} of sets that is closed under countable monotone unions and also under countable monotone intersections. Explicitly, this means M {\displaystyle M} {\displaystyle M} has the following properties:

  1. if A 1 , A 2 , … ∈ M {\displaystyle A_{1},A_{2},\ldots \in M} {\displaystyle A_{1},A_{2},\ldots \in M} and A 1 ⊆ A 2 ⊆ ⋯ {\displaystyle A_{1}\subseteq A_{2}\subseteq \cdots } {\displaystyle A_{1}\subseteq A_{2}\subseteq \cdots } then ⋃ i = 1 ∞ A i ∈ M , {\textstyle {\textstyle \bigcup \limits _{i=1}^{\infty }}A_{i}\in M,} {\textstyle {\textstyle \bigcup \limits _{i=1}^{\infty }}A_{i}\in M,} and
  2. if B 1 , B 2 , … ∈ M {\displaystyle B_{1},B_{2},\ldots \in M} {\displaystyle B_{1},B_{2},\ldots \in M} and B 1 ⊇ B 2 ⊇ ⋯ {\displaystyle B_{1}\supseteq B_{2}\supseteq \cdots } {\displaystyle B_{1}\supseteq B_{2}\supseteq \cdots } then ⋂ i = 1 ∞ B i ∈ M . {\textstyle {\textstyle \bigcap \limits _{i=1}^{\infty }}B_{i}\in M.} {\textstyle {\textstyle \bigcap \limits _{i=1}^{\infty }}B_{i}\in M.}

Monotone class theorem for sets

Monotone class theorem for sets—Let G {\displaystyle G} {\displaystyle G} be an algebra of sets and define M ( G ) {\displaystyle M(G)} {\displaystyle M(G)} to be the smallest monotone class containing G . {\displaystyle G.} {\displaystyle G.} Then M ( G ) {\displaystyle M(G)} {\displaystyle M(G)} is precisely the 𝜎-algebra generated by G {\displaystyle G} {\displaystyle G}; that is σ ( G ) = M ( G ) . {\displaystyle \sigma (G)=M(G).} {\displaystyle \sigma (G)=M(G).}

Monotone class theorem for functions

Monotone class theorem for functions—Let A {\displaystyle {\mathcal {A}}} {\displaystyle {\mathcal {A}}} be a π-system that contains Ω {\displaystyle \Omega \,} {\displaystyle \Omega \,} and let H {\displaystyle {\mathcal {H}}} {\displaystyle {\mathcal {H}}} be a collection of functions from Ω {\displaystyle \Omega } {\displaystyle \Omega } to R {\displaystyle \mathbb {R} } {\displaystyle \mathbb {R} } with the following properties:

  1. If A ∈ A {\displaystyle A\in {\mathcal {A}}} {\displaystyle A\in {\mathcal {A}}} then 1 A ∈ H {\displaystyle \mathbf {1} _{A}\in {\mathcal {H}}} {\displaystyle \mathbf {1} _{A}\in {\mathcal {H}}} where 1 A {\displaystyle \mathbf {1} _{A}} {\displaystyle \mathbf {1} _{A}} denotes the indicator function of A . {\displaystyle A.} {\displaystyle A.}
  2. If f , g ∈ H {\displaystyle f,g\in {\mathcal {H}}} {\displaystyle f,g\in {\mathcal {H}}} and c ∈ R {\displaystyle c\in \mathbb {R} } {\displaystyle c\in \mathbb {R} } then f + g {\displaystyle f+g} {\displaystyle f+g} and c f ∈ H . {\displaystyle cf\in {\mathcal {H}}.} {\displaystyle cf\in {\mathcal {H}}.}
  3. If f n ∈ H {\displaystyle f_{n}\in {\mathcal {H}}} {\displaystyle f_{n}\in {\mathcal {H}}} is a sequence of non-negative functions that increase to a bounded function f {\displaystyle f} {\displaystyle f} then f ∈ H . {\displaystyle f\in {\mathcal {H}}.} {\displaystyle f\in {\mathcal {H}}.}

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

Proof

The following argument originates in Rick Durrett's Probability: Theory and Examples.[1]

Proof

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

Results and applications

As a corollary, if G {\displaystyle G} {\displaystyle G} is a ring of sets, then the smallest monotone class containing it coincides with the 𝜎-ring of G . {\displaystyle G.} {\displaystyle G.}

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

See also

  • Dynkin system – Family closed under complements and countable disjoint unions
  • π-𝜆 theorem – Family closed under complements and countable disjoint unionsPages displaying short descriptions of redirect targets
  • π-system – Family of sets closed under intersection
  • σ-algebra – Algebraic structure of set algebra

Citations

  1. Durrett, Rick (2010). Probability: Theory and Examples (4th ed.). Cambridge University Press. p. 276. ISBN 978-0521765398.

References