Lemma on Measure of Increasing Sequences in X

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

Lemma: Let \mu be a measure defined on a \sigma-algebra X.

(a) If (E_n) is an increasing sequence in X, then

\mu (\bigcup_{n=1}^\infty E_n )=\lim \mu (E_n)

(b) If F_n) is a decreasing sequence in X and if \mu (F_1)<\infty, then

\mu (\bigcap_{n=1}^\infty F_n )=\lim \mu (F_n )

Note: An increasing sequence of sets (E_n) means that for all natural numbers n, E_n \subseteq E_{n+1}. A decreasing sequence means the opposite, i.e. E_n \supseteq E_{n+1}.

Proof: (Elaboration of the proof given in Bartle’s book)

(a) First we note that if \mu (E_n) = \infty for some n, then both sides of the equation are \infty, and inequality holds. Henceforth, we can just consider the case \mu (E_n)<\infty for all n.

Let A_1 = E_1 and A_n=E_n \setminus E_{n-1} for n>1. Then (A_n) is a disjoint sequence of sets in X such that

E_n=\bigcup_{j=1}^n A_j, \bigcup_{n=1}^\infty E_n = \bigcup_{n=1}^\infty A_n

Since \mu is countably additive,

\mu (\bigcap_{n=1}^\infty E_n) = \sum_{n=1}^\infty \mu (A_n) (since (A_n) is a disjoint sequence of sets)

=\lim_{m\to\infty} \sum_{n=1}^m \mu (A_n)

By an earlier lemma \mu (F\setminus E)=\mu (F)-\mu (E), we have that \mu (A_n)=\mu (E_n)-\mu (E_{n-1}) for n>1, so the finite series on the right side telescopes to become

\sum_{n=1}^m \mu (A_n)=\mu (E_m)

Thus, we indeed have proved (a).

For part (b),  let E_n=F_1 \setminus F_n, so that (E_n) is an increasing sequence of sets in X.

We can then apply the results of part (a).

\begin{aligned}    \mu (\bigcup_{n=1}^\infty E_n) &=\lim \mu (E_n)\\    &=\lim [\mu (F_1)-\mu (F_n)]\\    &=\mu (F_1) -\lim \mu (F_n)    \end{aligned}

Since we have \bigcup_{n=1}^\infty E_n =F_1 \setminus \bigcap_{n=1}^{\infty} F_n, it follows that

\mu (\bigcup_{n=1}^\infty E_n) =\mu (F_1)-\mu (\bigcap_{n=1}^\infty F_n)

Comparing the above two equations, we get our desired result, i.e. \mu (\bigcap_{n=1}^\infty F_n) = \lim \mu (F_n).

Reference:

The Elements of Integration and Lebesgue Measure

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