Monotone class theorem

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 {displaystyle G} is precisely the smallest -algebra containing  {displaystyle G.} It is used as a type of transfinite induction to prove many other theorems, such as Fubini's theorem.

Contents 1 Definition of a monotone class 2 Monotone class theorem for sets 3 Monotone class theorem for functions 3.1 Proof 4 Results and applications 5 See also 6 Citations 7 References Definition of a monotone class A monotone class is a family (i.e. class) {displaystyle M} of sets that is closed under countable monotone unions and also under countable monotone intersections. Explicitly, this means {displaystyle M} has the following properties: if {displaystyle A_{1},A_{2},ldots in M} and {displaystyle A_{1}subseteq A_{2}subseteq cdots } then {textstyle bigcup _{i=1}^{infty }A_{i}in M,} and if {displaystyle B_{1},B_{2},ldots in M} and {displaystyle B_{1}supseteq B_{2}supseteq cdots } then {textstyle bigcap _{i=1}^{infty }B_{i}in M.} Monotone class theorem for sets Monotone class theorem for sets — Let {displaystyle G} be an algebra of sets and define {displaystyle M(G)} to be the smallest monotone class containing {displaystyle G.} Then {displaystyle M(G)} is precisely the -algebra generated by {displaystyle G} ; that is, {displaystyle sigma (G)=M(G).} Monotone class theorem for functions Monotone class theorem for functions — Let {displaystyle {mathcal {A}}} be a π-system that contains {displaystyle Omega ,} and let {displaystyle {mathcal {H}}} be a collection of functions from {displaystyle Omega } to {displaystyle mathbb {R} } with the following properties: If {displaystyle Ain {mathcal {A}}} then {displaystyle mathbf {1} _{A}in {mathcal {H}}.} If {displaystyle f,gin {mathcal {H}}} and {displaystyle cin mathbb {R} } then {displaystyle f+g} and {displaystyle cfin {mathcal {H}}.} If {displaystyle f_{n}in {mathcal {H}}} is a sequence of non-negative functions that increase to a bounded function {displaystyle f} then {displaystyle fin {mathcal {H}}.} Then {displaystyle {mathcal {H}}} contains all bounded functions that are measurable with respect to {displaystyle sigma ({mathcal {A}}),} which is the sigma-algebra generated by {displaystyle {mathcal {A}}.} Proof The following argument originates in Rick Durrett's Probability: Theory and Examples.[1] Proof The assumption {displaystyle Omega ,in {mathcal {A}},} (2), and (3) imply that {displaystyle {mathcal {G}}=left{A:mathbf {1} _{A}in {mathcal {H}}right}} is a -system. By (1) and the π− theorem, {displaystyle sigma ({mathcal {A}})subset {mathcal {G}}.} Statement (2) implies that {displaystyle {mathcal {H}}} contains all simple functions, and then (3) implies that {displaystyle {mathcal {H}}} contains all bounded functions measurable with respect to {displaystyle sigma ({mathcal {A}}).} Results and applications As a corollary, if {displaystyle G} is a ring of sets, then the smallest monotone class containing it coincides with the sigma-ring of {displaystyle 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.

See also π- theorem π-system – Family of sets closed under intersection Dynkin system – Family closed under complements and countable disjoint unions Citations ^ Durrett, Rick (2010). Probability: Theory and Examples (4th ed.). Cambridge University Press. p. 276. ISBN 978-0521765398. References Durrett, Richard (2019). Probability: Theory and Examples (PDF). Cambridge Series in Statistical and Probabilistic Mathematics. Vol. 49 (5th ed.). Cambridge New York, NY: Cambridge University Press. ISBN 978-1-108-47368-2. OCLC 1100115281. Retrieved November 5, 2020. show Families {displaystyle {mathcal {F}}} of sets over {displaystyle Omega } Categories: Set familiesTheorems in measure theory

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