A Dynkin system, named after
Eugene Dynkin
Eugene Borisovich Dynkin (russian: link=no, Евгений Борисович Дынкин; 11 May 1924 – 14 November 2014) was a Soviet and American mathematician. He made contributions to the fields of probability and algebra, especially sem ...
is a
collection of
subsets of another universal
set
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*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
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
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speakin ...
.
A major application of -systems is the - theorem, see below.
Definition
Let
be a
nonempty
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
set, and let
be a
collection of subsets of
(that is,
is a subset of the
power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is post ...
of
). Then
is a Dynkin system if
#
#
is closed under
complements of subsets in supersets: if
and
then
#
is closed under countable increasing
union
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
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** ''Un ...
s: if
is an increasing sequence
[A sequence of sets is called if for all ] of sets in
then
It is easy to check
that any Dynkin system
satisfies:
- is closed under complements in : if then
* Taking shows that
- is closed under countable unions of
pairwise disjoint
In mathematics, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set.. For example, and are ''disjoint sets,'' while and are not disjoint. A c ...
sets: if is a sequence of pairwise disjoint
In mathematics, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set.. For example, and are ''disjoint sets,'' while and are not disjoint. A c ...
subsets in (meaning that for all ) then
* To be clear, this property also holds for finite sequences of pairwise disjoint sets (by letting for all ).
Conversely, it is easy to check that a family of sets that satisfy (4-6) is a Dynkin class.
For this reason, a small group of authors have adopted conditions 4-6 to define a Dynkin system as they are easier to verify.
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
of subsets of
there exists a unique Dynkin system denoted
which is minimal with respect to containing
That is, if
is any Dynkin system containing
then
is called the
For instance,
For another example, let
and
; then
Sierpiński-Dynkin's π-λ theorem
If
is a
-system and
is a Dynkin system with
then
In other words, the -algebra generated by
is contained in
Thus a Dynkin system contains a -system if and only if it contains the -algebra generated by that -system.
One application of Sierpiński-Dynkin's - theorem is the uniqueness of a measure that evaluates the length of an interval (known as the
Lebesgue measure):
Let
be the
unit interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis ...
,1with the Lebesgue measure on
Borel sets
In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named ...
. Let
be another
measure on
satisfying
and let
be the family of sets
such that
Let
and observe that
is closed under finite intersections, that
and that
is the -algebra generated by
It may be shown that
satisfies the above conditions for a Dynkin-system. From Sierpiński-Dynkin's - Theorem it follows that
in fact includes all of
which is equivalent to showing that the Lebesgue measure is unique on
Application to probability distributions
See also
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Notes
Proofs
References
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{{Measure theory
Families of sets
Lemmas
Probability theory