Find a concept
Start typing to search the mathematical index.
A set equipped with a sigma-algebra of measurable subsets.
A measurable space is a pair consisting of a set and a sigma-algebra on .
Measurable spaces are the domains and codomains for measurable functions; adding a measure produces a measure space.
A set is an object for which statements of the form (read “ is an element of ”) are meaningful, and whose identity is determined by extensionality: for sets ,
A sigma-algebra on a set is a nonempty collection such that if then , and whenever is a sequence in , the union lies in .
A measurable function between measurable spaces and is a function such that for every , the preimage lies in .
Measurability depends on the choice of sigma-algebras on domain and codomain; in particular, using the Borel sigma-algebra connects measurability to topology. For example, a continuous map between topological spaces is Borel measurable.
A measure on a measurable space is a function such that and for every pairwise disjoint sequence in ,
A measure assigns “sizes” to measurable sets and turns into a measure space. Sets of measure zero are null sets, and they control notions like almost everywhere.
A measure space is a triple where is a measurable space and is a measure on .
Measure spaces provide the basic setting for integration, convergence theorems, and statements that hold almost everywhere rather than pointwise.
A Borel sigma-algebra on a topological space is the sigma-algebra generated by the open sets of , i.e. the smallest sigma-algebra that contains every open subset of .
A power set of a set is the set