# 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 $\mathrm{(}X\mathrm{,}\mathrm{F}\mathrm{)}$ 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 $\mathcal{F}$ 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 $(\mathrm{\Omega},\mathcal{F})$, 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 ${\pi}_{\mathrm{\Omega}}:\mathrm{\Omega}\times {\mathbb{R}}_{+}\to \mathrm{\Omega}$ 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 |