Sigma Algebra

• For a sequence of sets , we define the infimum and supremum as follows.

  The limit supremum and infimum are defined as

  • Intuition: The limit supremum contains all points that are in infinitely many of these sets.
  • Intuition: The limit infimum contains all points that are in all but finitely many of the sets.

• If for some sequence of subsets

  Then is the limit of denoted

• (Resnick 1.3.1) Let be a sequence of subsets of . We have the following

  For some subsequence depending on . We write this as

  Wee also have

• The following holds

  Using De Morgan’s we also get

• A sequence of sets is monotone non-decreasing if . It is monotone increasing if . We notate to mean monotone non-decreasing and to mean strictly non-decreasing. We also use to mean monotone non-increasing and to mean strictly non-increasing

• (Resnick 1.4.1) Suppose is a monotone sequence of subsets. Then

  Since for any sequences , we have

  It follows that

• The indicator function satisfies the following

  • We have the following inequality
    If the sequence is mutually disjoint, then equality holds

• Let be a set and . Then, is a -algebra if it satisfies the following properties

• By De Morgan’s Law, a -algebra is also closed under intersection. In fact, *-algebras are closed under countable union, intersection, and complementation

• (Resnick Corr.1.6.1) The intersection of -algebras is a -algebra

• Let . The -field generated by denoted is a -algebra satisfy ing

  • If is some other -algebra containing , then . Thus is the minimal -algebra over

• (Resnick 1.6.1) Given a class , there is a unique minimal -algebra containing .

  • Proof: Let be the set of -algebras containing . It is easy to show that so at least one -algebra exists. Then the intersection is also a -algebra and clearly so .

• (Resnick 1.8.1) Let .

  • If is a -algebra of subsets of , then is a -algebra of subsets of . We denote this as
  • If and . Then
  • (Resnick Corr.1.8.1) If then

• Let be a topological space where we have a notion of “open sets”.

  Then the Borel -Algebra is defined as the -algebra generated by the set of open sets .

Links §

• A Probability Path by Resnick
• Set Theory - The -algebra operates on sets.
• Real Analysis - -algebra has Real Analysis analogues