Section 2.5

Countable Additivity and Continuity of Measure, and the Borel-Cantelli Lemma

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Problem 2.22

Show that if $E_{1}$ and $E_{2}$ are measurable sets of finite measure, then

m\left(E_{1} \cup E_{2}\right)-m\left(E_{1} \cap E_{2}\right) = m\left(E_{1} \sim E_{2}\right)+m\left(E_{2} \sim E_{1}\right)

Problem 2.23

Show that the assumption that $m\left(B_{1}\right) < \infty$ is necessary in part (ii) of the theorem regarding continuity of measure.

Problem 2.24

Let $\left\{ E_{k}\right\}_{k = 1}^{\infty }$ be a countable, disjoint collection of measurable sets. Prove that for any set $A$ ,

m^{*}\left(A \cap \bigcup _{k = 1}^{\infty } E_{k}\right) = \sum _{k = 1}^{\infty } m^{*}\left(A \cap E_{k}\right)

Problem 2.25

(i)(ii)

Let $\mathcal{M}^{\prime }$ be any $\sigma$ -algebra of subsets of $\mathbb {R}$ and $m^{\prime }$ a set-function on $\mathcal{M}^{\prime }$ that takes values in $[0, \infty )$ , is countably additive, and such that $m^{\prime }(\emptyset ) = 0$ .

(i)

Show that $m^{\prime }$ is finitely additive, monotone, countably monotone, and possesses the excision property.

(ii)

Show that $m^{\prime }$ possesses the same continuity properties as Lebesgue measure.

Problem 2.26

Show that the continuity of measure together with the finite additivity of measure are equivalent to the countable additivity of measure.

Problem 2.27

Let $\left\{ E_{k}\right\}_{k = 1}^{\infty }$ be a collection of measurable subsets of $[a, b]$ , and assume there is a $\delta > 0$ such that $m\left(E_{k}\right) \geq \delta$ for each $k$ . Show that there is an $x \in [a, b]$ that belongs to infinitely many such $E_{k}$ 's.