(Continued from https://mathtuition88.com/2015/06/20/what-is-a-measure-measure-theory/)

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. .

Reference:

The Elements of Integration and Lebesgue Measure

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