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Definition
A measurable family of probability measures indexed by another measurable space.
A probability kernel from to assigns a probability measure on to each , such that is measurable for every .
Given probability on , the formula defines a probability measure on . A disintegration uses such a kernel to describe conditional probabilities on fibers of a measurable map.
A probability measure is a function defined on a sigma-algebra of subsets of a set such that , for every pairwise disjoint sequence , and .
A probability measure is a special case of a measure and is the key ingredient in a probability space; it assigns probabilities to events (measurable sets).
Let be measurable between standard Borel spaces, let be a Borel probability on , and let . There is a probability kernel , unique for -almost every , such that almost everywhere and
for every bounded Borel . It is called the disintegration of over .