measurable projection theorem
The projection of a measurable set from the product of two measurable spaces need not itself be measurable. See a Lebesgue measurable but non-Borel set for an example. However, the following result can be shown. The notation refers to the product -algebra (http://planetmath.org/ProductSigmaAlgebra).
In particular, if is universally complete then the projection of will be in , and this applies to all complete -finite (http://planetmath.org/SigmaFinite) measure spaces . For example, the projection of any Borel set in onto is Lebesgue measurable.
The theorem is a direct consequence of the properties of analytic sets (http://planetmath.org/AnalyticSet2), following from the result that projections of analytic sets are analytic and the fact that analytic sets are universally measurable (http://planetmath.org/MeasurabilityOfAnalyticSets). Note, however, that the theorem itself does not refer at all to the concept of analytic sets.
The measurable projection theorem has important applications to the theory of continuous-time stochastic processes. For example, the début theorem, which says that the first time at which a progressively measurable stochastic process enters a given measurable set is a stopping time, follows easily. Also, if is a jointly measurable process defined on a measurable space , then the maximum process will be universally measurable since,
|Title||measurable projection theorem|
|Date of creation||2013-03-22 18:48:04|
|Last modified on||2013-03-22 18:48:04|
|Last modified by||gel (22282)|