Hierarchy of Borel

Definition of the whole of Borel

A algebra on a

X

unit is a collection

\ mathcal {has}

subsets of

X

checking the following conditions:

$X \ in \ mathcal {has}$
If $has \ in \ mathcal {has}$ , then $A^c \ in \ mathcal {has}$
Any union finished of elements of $\ mathcal {has}$ belongs to $\ mathcal {has}$ .

To notice that $\ emptyset \ in \ mathcal {has}$ for any algebra $\ mathcal {has}$ . An algebra $\ mathcal {has}$ such as any countable union of elements of $\ mathcal {has}$ belongs to $\ mathcal {has}$ is called a $\ sigma$ -algèbre .

It is easy to see that the intersection of a not-vacuum family of $\ sigma$ -algèbres on $X$ is a $\ sigma$ -algèbre. This observation allows the following definition. That is to say $\ mathcal {G}$ a family of subsets of $X$ . That is to say $\ mathcal {S}$ the whole of the $\ sigma$ -algèbres on $X$ container $\ mathcal {G}$ . To note that $\ mathcal {S}$ is not-vacuum bus the $\ sigma$ -algèbre $\ mathcal {P} (X)$ contains trivialement $\ mathcal {G}$ . One calls $\ sigma$ -algèbre generated by $\ mathcal {G}$ } the intersection of all the members of $\ mathcal {S}$ .

That is to say $(X, \ mathcal {T})$ a topological space métrisable. One calls $\ sigma$ -algèbre of Boréliens on X the $\ sigma$ -algèbre generated by $\ mathcal {T}$ . It is noted $\ mathcal {B} _X$ . A member of the $\ sigma$ -algèbre of Boréliens is called a borélien or together of Borel .

Hierarchy of Boréliens

That is to say $\ mathcal {F}$ a family of subsets of a $X$ unit. One notes $\ mathcal {F} _ {\ delta}$ the whole of the countable unions of elements of $\ mathcal {F}$ :

$\ mathcal {F} _ {\ sigma} = \ {\ bigcup_ {N \ in \ NR} A_n \ mid (A_n) _ {N \ in \ NR} \ subset \ mathcal {F} \}.$

One also notes by $\ mathcal {F} _ {\ sigma}$ the whole of the countable intersections of $\ mathcal {F}$ :

$\ mathcal {F} _ {\ delta} = \ {\ bigcap_ {N \ in \ NR} A_n \ mid (A_n) _ {N \ in \ NR} \ subset \ mathcal {F} \}.$

One désingne finally by $\ neg \ mathcal {F}$ the whole of the complements in $X$ of the elements of $\ mathcal {F}$ :

$\ neg \ mathcal {F} = \ {has \ subset X \ mid X \ backslash has \ subset \ mathcal {F} \}.$

That is to say a topological space $(X, \ mathcal {T})$ . Let us note in the following way open and closed $X$ :

$\ Sigma_1^0 (X) = \ mathcal {T},$

$\ Pi_1^0 (X) = \ neg \ mathcal {T}.$

Then for each ordinal $\ alpha$ , $1 < \ alpha < \ omega_1$ , one then defines the families of following whole per transfinite induction:

$\ Sigma_ {\ alpha} ^0 (X) = (\ bigcup_ {\ beta< \ alpha} \ Pi_ {\ beta} ^0 (X))_ {\ sigma},$

$\ Pi_ {\ alpha} ^0 (X) = (\ bigcup_ {\ beta< \ alpha} \ Sigma_ {\ beta} ^0 (X))_ {\ delta}.$

Finally for each ordinal $\ alpha$ , $1 \ Leq \ alpha < \ omega_1$ , one defines:

$\ Delta_ {\ alpha} ^0 = \ Sigma_ {\ alpha} ^0 \ bigcup \ Pi_ {\ alpha} ^0.$

Let us note that $\ Delta_ {\ alpha} ^0$ is the family of the whole of $X$ which at the same time open and is closed for topology $\ mathcal {T}$ . If there is no ambiguity, or if a result is valid for any topological space $(X, \ mathcal {T})$ , one notes sometimes $\ Sigma_ {\ alpha} ^0$ , $\ Pi_ {\ alpha} ^0$ and $\ Delta_ {\ alpha} ^0$ instead of $\ Sigma_ {\ alpha} ^0 (X)$ , $\ Pi_ {\ alpha} ^0 (X)$ and $\ Delta_ {\ alpha} ^0 (X)$ . The families $\ Sigma_ {\ alpha} ^0$ , $\ Pi_ {\ alpha} ^0$ and $\ Delta_ {\ alpha} ^0$ are called the additive classes , multiplicative and ambiguous . These families of units check the following elementary properties.

the additive classes are closed by countable unions, and the multiplicative classes are closed by countable intersections.
For very ordinal $\ alpha$ , $1 \ Leq \ alpha < \ omega_1$ , $\ Pi_ {\ alpha} ^0 = \ neg \ Sigma_ {\ alpha} ^0$ , or in an equivalent way $\ Sigma_ {\ alpha} ^0 =$

\ neg \ Pi_ {\ alpha} ^0.

For very ordinal $\ alpha$ , $1 \ Leq \ alpha < \ omega_1$ , $\ Delta_ {\ alpha} ^0$ is an algebra.

It is shown whereas:

$\ mathcal {B} _X = \ bigcup_ {1 \ Leq \ alpha < \ omega_1} \ Sigma_ {\ alpha} ^0 = \ bigcup_ {1 \ Leq \ alpha < \ omega_1} \ Pi_ {\ alpha} ^0 = \ bigcup_ {1 \ Leq \ alpha < \ omega_1} \ Delta_ {\ alpha} ^0.$

Random links: