Little lemma on nonmeasurable functions

Let f be a nonmeasurable function ("non-measurable random variable") on a probability space. Then there are measurable functions f^L and f^U such that f^L≤f≤g^U everywhere, and such that for every pair of measurable functions a and b such that a≤f≤b we have a≤f^L and g^U≤b almost surely.

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