# measurable projection theorem

###### 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.

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