Remark 2. The theorem remains true if its assumptions hold formula_47-almost everywhere. In other words, it is enough that there is a null set formula_7 such that the sequence formula_49 non-decreases for every formula_50 To see why this is true, we start with an observation that allowing the sequence formula_51 to pointwise non-decrease almost everywhere causes its pointwise limit formula_41 to be undefined on some null set formula_7. On that null set, formula_41 may then be defined arbitrarily, e.g. as zero, or in any other way that preserves measurability. To see why this will not affect the outcome of the theorem, note that since formula_55 we have, for every formula_56