measurable projection theorem

The projection of a measurable setMathworldPlanetmath from the productPlanetmathPlanetmathPlanetmath X×Y of two measurable spacesMathworldPlanetmath 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 σ-algebra (


Let (X,F) be a measurable space and Y be a Polish spaceMathworldPlanetmath with Borel σ-algebra B. Then the projection ( of any SF×B onto X is universally measurable.

In particular, if is universally complete then the projection of S will be in , and this applies to all completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath σ-finite ( measure spacesMathworldPlanetmath (X,,μ). For example, the projection of any Borel set in n onto is Lebesgue measurable.

The theorem is a direct consequence of the properties of analytic setsMathworldPlanetmath (, following from the result that projections of analytic sets are analytic and the fact that analytic sets are universally measurable ( 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 processMathworldPlanetmath enters a given measurable set is a stopping time, follows easily. Also, if (Xt)t+ is a jointly measurable process defined on a measurable space (Ω,), then the maximum process Xt*=supstXs will be universally measurable since,


is universally measurable, where πΩ:Ω×+Ω is the projection map. Furthermore, this also shows that the topologyMathworldPlanetmath of ucp convergence is well defined on the space of jointly measurable processes.

Title measurable projection theorem
Canonical name MeasurableProjectionTheorem
Date of creation 2013-03-22 18:48:04
Last modified on 2013-03-22 18:48:04
Owner gel (22282)
Last modified by gel (22282)
Numerical id 8
Author gel (22282)
Entry type Theorem
Classification msc 28A05
Classification msc 60G07
Related topic ProjectionsOfAnalyticSetsAreAnalytic