Algebra of sets

Set algebra in set theory is a nonempty system of subsets closed with respect to complement (difference) and union (sum) operations.

Content

Definition

Family ${\mathfrak {A}}\subset 2^{X}$ ${\ displaystyle {\ mathfrak {A}} \ subset 2 ^ {X}}$ subsets of the set $X$ ${\ displaystyle X}$ (here $2^{X}$ ${\ displaystyle 2 ^ {X}}$ - Boolean ) is called an algebra if it satisfies the following properties:

$\varnothing \in {\mathfrak {A}}.$ ${\ displaystyle \ varnothing \ in {\ mathfrak {A}}.}$
If the set $A\in {\mathfrak {A}}$ ${\ displaystyle A \ in {\ mathfrak {A}}}$ then its addition $X\setminus A\in {\mathfrak {A}}.$ ${\ displaystyle X \ setminus A \ in {\ mathfrak {A}}.}$
The union of two sets $A,B\in {\mathfrak {A}}$ ${\ displaystyle A, B \ in {\ mathfrak {A}}}$ also belongs ${\mathfrak {A}}.$ ${\ displaystyle {\ mathfrak {A}}.}$

Remarks

By definition, if an algebra contains a set $A$ ${\ displaystyle A}$ , then it contains its addition. Unification $A$ ${\ displaystyle A}$ with its complement is the original set $X$ ${\ displaystyle X}$ . Complement to the set $X$ ${\ displaystyle X}$ is an empty set. This means that many $X$ ${\ displaystyle X}$ and the empty set is contained in algebra by definition.
Due to the properties of operations on sets, the set algebra is also closed with respect to intersection and symmetric difference .
Set algebra is a special case of unit algebra , where the operation of "multiplication" is the intersection of sets, and the operation of "addition" is a symmetric difference.
If the original set $X$ ${\ displaystyle X}$ is the space of elementary events , then the algebra ${\mathfrak {A}}$ ${\ displaystyle {\ mathfrak {A}}}$ called the algebra of events - the key concept of probability theory and related mathematical disciplines, which has a unique interpretation and plays an independent role in mathematics.

Event Algebra

Algebra of events (in probability theory ) - the algebra of subsets of the space of elementary events $\Omega$ ${\ displaystyle \ Omega}$ whose elements are elementary events .

As befits a set algebra, an event algebra contains an impossible event (an empty set ) and is closed with respect to set-theoretic operations performed in a finite number. It is enough to demand that the algebra of events be closed with respect to two operations, for example, intersection and complement , from which its closure with respect to any other set-theoretic operations will immediately follow. The event algebra , closed with respect to a countable number of set-theoretic operations, is called the sigma-algebra of events .

In probability theory, the following algebras and sigma-algebras of events are found:

algebra of finite subsets $\Omega$ ${\ displaystyle \ Omega}$ ;
sigma algebra of countable subsets $\Omega$ ${\ displaystyle \ Omega}$ ;
subset algebra ${\mathbb {R} }^{n}$ ${\ displaystyle {\ mathbb {R}} ^ {n}}$ formed by finite unions of intervals ;
sigma-algebra of Borel subsets of a topological space $\Omega$ ${\ displaystyle \ Omega}$ , that is, the smallest sigma-algebra containing all open subsets $\Omega$ ${\ displaystyle \ Omega}$ ;
the algebra of cylinders in the space of functions; and the sigma-algebra generated by them.

Event $A+B$ ${\ displaystyle A + B}$ or $A\cup B$ ${\ displaystyle A \ cup B}$ , is that of the two events $A$ ${\ displaystyle A}$ and $B$ ${\ displaystyle B}$ at least one thing happens, called the sum of the events $A$ ${\ displaystyle A}$ and $B$ ${\ displaystyle B}$ .

A probability space is an algebra of events with a given probability function $\mathbb {P}$ ${\ displaystyle \ mathbb {P}}$ , i.e., a sigma-additive finite measure , the domain of which is the algebra of events, and $\mathbb {P} (\Omega )=1$ ${\ displaystyle \ mathbb {P} (\ Omega) = 1}$ .

Any sigma additive probability on an event algebra uniquely extends to a sigma additive probability defined on an sigma event algebra generated by a given event algebra .

Literature

Kolmogorov A.N. , Fomin S.V. Elements of function theory and functional analysis. - ed. fourth, revised. - M .: Science , 1976 . - 544 p.