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