Distributive law for countable union and intersection
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Let X be a nonempty set. A collection S of subsets of X is called a semiring if it satisfies the following properties:
The empty set belongs to S; that is ∅∈S.
If A,B∈S; then A∩B∈S; that is, S is closed under finite intersections.
The set difference of any two sets of S can be written as a finite union of pair-wise disjoint members of S. That is, for every A,B∈S; there exist C1,...,Cn∈S(depending on A and B) such that A∖B=∪ni=1Ci and Ci∩Cj=∅ if i≠j.
Now, let S be a semiring of subsets of X. A subset A of X is called a σ-set with respect to S (or simply a σ-set) if there exists a disjoint sequence {An} ofS such that A=∪∞n=1An. One can show easily that for every sequence {An} of S, the set A=∪∞n=1Anis a σ-set.
I would like to prove that finite intersections of σ-sets is a σ-set. For this purpose, suppose A,B are σ-sets then A=∪∞i=1Ci , B=∪∞j=1Dj.
A∩B=(∪∞i=1Ci)∩(∪∞j=1Dj)
In this step I don't know am I allowed to use distributive law for infinitely many sets? Or the law holds only for finitely many sets?
If it holds only for finitely many sets how do I prove that finite intersections of σ-sets is a σ-set?
The empty set belongs to S; that is ∅∈S.
If A,B∈S; then A∩B∈S; that is, S is closed under finite intersections.
The set difference of any two sets of S can be written as a finite union of pair-wise disjoint members of S. That is, for every A,B∈S; there exist C1,...,Cn∈S(depending on A and B) such that A∖B=∪ni=1Ci and Ci∩Cj=∅ if i≠j.
Now, let S be a semiring of subsets of X. A subset A of X is called a σ-set with respect to S (or simply a σ-set) if there exists a disjoint sequence {An} ofS such that A=∪∞n=1An. One can show easily that for every sequence {An} of S, the set A=∪∞n=1Anis a σ-set.
I would like to prove that finite intersections of σ-sets is a σ-set. For this purpose, suppose A,B are σ-sets then A=∪∞i=1Ci , B=∪∞j=1Dj.
A∩B=(∪∞i=1Ci)∩(∪∞j=1Dj)
In this step I don't know am I allowed to use distributive law for infinitely many sets? Or the law holds only for finitely many sets?
If it holds only for finitely many sets how do I prove that finite intersections of σ-sets is a σ-set?
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