measurable section theorem
Consider a Cartesian product with projection map . Then, a cross section of a subset is a map satisfying . The following theorem states that such cross sections can be chosen in a measurable (http://planetmath.org/MeasurableFunctions) way. We use the notation for the product -algebra (http://planetmath.org/ProductSigmaAlgebra) in order to distinguish it from the product paving .
For any -analytic set , then and there is a measurable map such that for all .
The existence of a map satisfying is no surprise, and indeed must exist by a simple application of the axiom of choice. The main force of the theorem is that 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 universal completeness assumption for or the requirement that be a Polish space is dropped.
Consider stochastic processes and defined on a probability space . Suppose that they were not equivalent so that, with positive probability, there is a time at which they differ. Then the projection of the set of times has positive probability. Consequently, the measurable section theorem guarantees the existence of a random time for which with positive probability.
Let and be jointly measurable stochastic processes defined on the probability space . If
for all random times , then and 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 be universally complete can be dropped from the statement, due to the fact that any random variable on the completion of a probability space can be replaced by a random variable satisfying .
|Title||measurable section theorem|
|Date of creation||2013-03-22 18:48:21|
|Last modified on||2013-03-22 18:48:21|
|Last modified by||gel (22282)|
|Synonym||measurable cross section theorem|