measurable section theorem

Consider a Cartesian product X×Y with projection map πX:X×YX. Then, a cross section of a subset SX×Y is a map τ:πX(S)Y satisfying (x,τ(x))S. The following theorem states that such cross sections can be chosen in a measurable ( way. We use the notation for the productPlanetmathPlanetmathPlanetmath σ-algebra ( in order to distinguish it from the product paving ×.


Let (X,F) be a universally complete measurable spaceMathworldPlanetmathPlanetmath (, Y be a Polish spaceMathworldPlanetmath with Borel σ-algebra ( B, and πX:X×YX be the projection map.

For any FB-analytic setMathworldPlanetmath S, then πX(S)F and there is a measurable map τ:πX(S)Y such that (x,τ(x))S for all xπX(S).

The existence of a map satisfying (x,τ(x))S is no surprise, and indeed must exist by a simple application of the axiom of choiceMathworldPlanetmath. The main force of the theorem is that τ may be taken to be measurable, which is not something that could be achieved by such basic use of choice. In fact, the result fails to be true if either the universalPlanetmathPlanetmath completeness assumptionPlanetmathPlanetmath for (X,) or the requirement that Y be a Polish space is dropped.

This result has important applications to continuous-time stochastic processes, where the space Y is taken to be the time index setMathworldPlanetmathPlanetmath, usually the set + of nonnegative real numbers.

Consider stochastic processesMathworldPlanetmath (Ut)t+ and (Vt)t+ defined on a probability spaceMathworldPlanetmath (Ω,,). Suppose that they were not equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath so that, with positive probability, there is a time t at which they differ. Then the projection πΩ(S) of the set of times S={(t,ω):Ut(ω)Vt(ω)} has positive probability. Consequently, the measurable section theorem guarantees the existence of a random time τ for which UτVτ with positive probability.


Let (Ut)tR+ and (Vt)tR+ be jointly measurable stochastic processes defined on the probability space (Ω,F,P). If


for all random times τ:ΩR+, then U and V are equivalent processes.

So, continuous-time stochastic processes are fully determined up to equivalence by their values, neglecting zero probability sets, at random times. Note that the condition that (Ω,) be universally complete can be dropped from the statement, due to the fact that any random variableMathworldPlanetmath τ on the completionPlanetmathPlanetmath of a probability space can be replaced by a random variable τ satisfying (τ=τ)=1.

More useful statements which apply to adapted processes, and allow τ to be either a stopping time or a predictable stopping time, are given by the optional and predictable sectionPlanetmathPlanetmath theorems.

Title measurable section theorem
Canonical name MeasurableSectionTheorem
Date of creation 2013-03-22 18:48:21
Last modified on 2013-03-22 18:48:21
Owner gel (22282)
Last modified by gel (22282)
Numerical id 7
Author gel (22282)
Entry type Theorem
Classification msc 28A05
Synonym measurable cross section theorem