real line, then there exist sets for which no size exists, for example, the Vitali sets. For this reason, one considers instead a smaller collection of privileged subsets of These subsets will be called the measurable sets. They are closed under operations that one would expect for measurable sets, that is, the complement of a measurable set is a measurable set and the countable union of measurable sets is a measurable set. Non-empty collections of sets with these properties are called σ-algebras. Limits of sets Many uses of measure, such as the probability concept of almost sure convergence, involve limits of sequences of sets. For this, closure under countable unions and intersections is paramount. Set limits are defined as follows on σ-algebras. The or of a sequence of subsets of is It consists of all points that are in infinitely many of these sets (or equivalently, that are in many of them). That is, if and only if there exists an infinite subsequence (where ) of sets that all contain that is, such that The or of a sequence of subsets of is It consists of all points that are in all but finitely many of these sets (or equivalently, that are in all of them). That is, if and only if there exists an index such that all contain that is, such that The inner limit is always a subset of the outer limit: If these two sets are equal then their limit exists and is equal to this common set: Sub σ-algebras In much of probability, especially when conditional expectation is involved, one is concerned with sets that represent only part of all the possible information that can be observed. This partial information can be characterized with a smaller σ-algebra which is a subset of the principal σ-algebra; it consists of the collection of subsets relevant only to and determined only by the partial information. A simple example suffices to illustrate this idea. Imagine you and another person are betting on a game that involves flipping a coin repeatedly and observing whether it comes up Heads () or Tails (). Since you and your opponent are each infinitely wealthy, there is no limit to how long the game can last. This means the sample space Ω must consist of all possible infinite sequences of or However, after flips of the coin, you may want to determine or revise