Lemma: Let be a measure defined on a -algebra X.
(a) If () is an increasing sequence in X, then
(b) If ) is a decreasing sequence in X and if , then
Note: An increasing sequence of sets () means that for all natural numbers n, . A decreasing sequence means the opposite, i.e. .
Proof: (Elaboration of the proof given in Bartle’s book)
(a) First we note that if for some n, then both sides of the equation are , and inequality holds. Henceforth, we can just consider the case for all n.
Let and for n>1. Then is a disjoint sequence of sets in X such that
Since is countably additive,
(since () is a disjoint sequence of sets)
By an earlier lemma , we have that for n>1, so the finite series on the right side telescopes to become
Thus, we indeed have proved (a).
For part (b), let , so that is an increasing sequence of sets in X.
We can then apply the results of part (a).
Since we have , it follows that
Comparing the above two equations, we get our desired result, i.e. .