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.