measurable projection theorem

The projection of a measurable set from the product $X\times Y$ 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 $ℱ\times ℬ$ refers to the product $\sigma$-algebra (http://planetmath.org/ProductSigmaAlgebra).

Theorem.

Let $\mathrm{(}X\mathrm{,}\mathrm{F}\mathrm{)}$ be a measurable space and $Y$ be a Polish space with Borel $\sigma$-algebra $\mathrm{B}$. Then the projection (http://planetmath.org/ProjectionMap) of any $S\mathrm{\in }\mathrm{F}\mathrm{\times }\mathrm{B}$ onto $X$ is universally measurable.

In particular, if $ℱ$ is universally complete then the projection of $S$ will be in $ℱ$, and this applies to all complete $\sigma$-finite (http://planetmath.org/SigmaFinite) measure spaces $(X,ℱ,\mu )$. For example, the projection of any Borel set in ${ℝ}^n$ onto $ℝ$ 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 ${(X_t)}_{t\in {ℝ}_{+}}$ is a jointly measurable process defined on a measurable space $(\mathrm{\Omega },ℱ)$, then the maximum process $X_t^{*}={sup}_{s\le t}{X}_s$ will be universally measurable since,

$$\{\omega \in \mathrm{\Omega }:X_t^{*}>K\}={\pi }_{\mathrm{\Omega }}\left(\{(s,\omega ):s\le t,X_s>K\}\right)$$

is universally measurable, where ${\pi }_{\mathrm{\Omega }}:\mathrm{\Omega }\times {ℝ}_{+}\to \mathrm{\Omega }$ is the projection map. Furthermore, this also shows that the topology 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