# 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 $\mathcal{F}\times\mathcal{B}$ refers to the product $\sigma$-algebra (http://planetmath.org/ProductSigmaAlgebra).

###### Theorem.

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

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

 $\left\{\omega\in\Omega\colon X^{*}_{t}>K\right\}=\pi_{\Omega}\left(\left\{(s,% \omega)\colon s\leq t,\ X_{s}>K\right\}\right)$

is universally measurable, where $\pi_{\Omega}\colon\Omega\times\mathbb{R}_{+}\to\Omega$ 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 MeasurableProjectionTheorem 2013-03-22 18:48:04 2013-03-22 18:48:04 gel (22282) gel (22282) 8 gel (22282) Theorem msc 28A05 msc 60G07 ProjectionsOfAnalyticSetsAreAnalytic