the property under consideration while, on the other hand, showing that the collection of all subsets with the property is a Dynkin system can also be straightforward. Dynkin's π-λ Theorem then implies that all sets in enjoy the property, avoiding the task of checking it for an arbitrary set in One of the most fundamental uses of the π-λ theorem is to show equivalence of separately defined measures or integrals. For example, it is used to equate a probability for a random variable with the Lebesgue-Stieltjes integral typically associated with computing the probability: for all in the Borel σ-algebra on where is the cumulative distribution function for defined on while is a probability measure, defined on a σ-algebra of subsets of some sample space Combining σ-algebras Suppose is a collection of σ-algebras on a space Meet The intersection of a collection of σ-algebras is a σ-algebra. To emphasize its character as a σ-algebra, it often is denoted by: Sketch of Proof: Let denote the intersection. Since is in every is not empty. Closure under complement and countable unions for every implies the same must be true for Therefore, is a σ-algebra. Join The union of a collection of σ-algebras is not generally a σ-algebra, or even an algebra, but it generates a σ-algebra known as the join which typically is denoted A π-system that generates the join is Sketch of Proof: By the case it is seen that each so This implies by the definition of a σ-algebra generated by a collection of subsets. On the other hand, which, by Dynkin's π-λ theorem, implies σ-algebras for subspaces Suppose is a subset of and let be a measurable space. The collection is a σ-algebra of subsets of Suppose is a measurable space. The collection is a σ-algebra of subsets of Relation to σ-ring A σ-algebra is just a σ-ring that contains the universal set A σ-ring need not be a σ-algebra, as for example measurable subsets of zero Lebesgue measure in the real line are a σ-ring, but not a σ-algebra since the real line has infinite measure and thus cannot be obtained by their countable union. If, instead of zero measure, one takes measurable subsets of finite Lebesgue measure, those are a ring but not a σ-ring, since the real line can be obtained by their countable union yet its measure is not finite. Typographic note σ-algebras are sometimes denoted