Dynkin system
A Dynkin system, named after Eugene Dynkin, is a collection of subsets of another universal set formula_1 satisfying a set of axioms weaker than those of -algebra. Dynkin systems are sometimes referred to as -systems (Dynkin himself used this term) or d-system. These set families have applications in measure theory and probability. A major application of -systems is the - theorem, see below. Definitions. Let Ω be a nonempty set, and let formula_2 be a collection of subsets of formula_1 (that is, formula_2 is a subset of the power set of formula_1). Then formula_2 is a Dynkin system  if Equivalently, formula_2 is a Dynkin system if The second definition is generally preferred as it usually is easier to check. An important fact is that a Dynkin system which is also a -system (that is, closed under finite intersections) is a -algebra. This can be verified by noting that conditions 2 and 3 together with closure under finite intersections imply closure under countable unions. Given any collection formula_30 of subsets of formula_31 there exists a unique Dynkin system denoted formula_32 which is minimal with respect to containing formula_33 That is, if formula_34 is any Dynkin system containing formula_35 then  formula_36 formula_32 is called the For instance, formula_38 For another example, let formula_39 and formula_40; then formula_41 Dynkin's π-λ theorem. If formula_42 is a -system and formula_2 is a Dynkin system with formula_44 then formula_45 In other words, the -algebra generated by formula_42 is contained in formula_47 One application of Dynkin's - theorem is the uniqueness of a measure that evaluates the length of an interval (known as the Lebesgue measure): Let formula_48 be the unit interval [0,1] with the Lebesgue measure on Borel sets. Let formula_49 be another measure on formula_1 satisfying formula_51 and let formula_2 be  the family of sets formula_53 such that formula_54 Let formula_55 and observe that formula_56 is closed under finite intersections, that formula_57 and that formula_58 is the -algebra generated by formula_59 It may be shown that formula_2 satisfies the above conditions for a Dynkin-system. From Dynkin's - Theorem it follows that formula_2 in fact includes all of formula_62 which is equivalent to showing that the Lebesgue measure is unique on formula_63