Wednesday, July 25, 2012

Little lemma on nonmeasurable functions

Let f be a nonmeasurable function ("non-measurable random variable") on a probability space. Then there are measurable functions fL and fU such that fLfgU everywhere, and such that for every pair of measurable functions a and b such that afb we have afL and gUb almost surely.

I.e., the function fL is the unique-up-to-null-sets largest measurable function less than or equal to f, and fU is the unique-up-to-null-sets smallest measurable function greater than or equal to g. Basically, fL and fU encode all the probabilistic information available about f.

1 comment:

Alexander R Pruss said...

Reference for lemma:
Lemmas 1.2.2 in A. van der Vaart and J. Wellner. Weak Convergence and Empirical Processes: With Applications to Statistics, New York: Springer, 1996