In measure theory and probability theory, Bierlein's measure extension theorem is a result about the extensibility of a probability measure. It states that a probability measure can be extended to the -algebra generated by the union of the underlying -algebra and an arbitrary family of disjoint subsets of the space. It has particular importance when dealing with infinite-dimensional spaces.
The statement of Bierlein's measure extension theorem is
Let be a probability space and a family of disjoint sets in for some index set . Then there is an extension of to .