Math Problems Vault

Problem 10.1

Show that if conditions (i) and (iii) in the definition of measure hold, and if $\mu (A)<\infty$ for some $A \in \mathscr {F}$ , then condition (ii) holds.

?

Problem 10.2

On the $\sigma$ -field of all subsets of $\Omega =\left\{ 1,2, \ldots \right\}$ put $\mu (A)=\Sigma_{k \in A^{2}} 2^{-k}$ if $A$ is finite and $\mu (A)=\infty$ otherwise. Is $\mu$ finitely additive? Countably additive?

?

Problem 10.3

(a)(b)

?

(a)

In connection with Theorem 10.2(ii), show that if $A_{n} \downarrow A$ and $\mu \left(A_{k}\right)<\infty$ for some $k$ , then $\mu \left(A_{n}\right) \downarrow \mu (A)$ .

(b)

Find an example in which $A_{n} \downarrow A, \mu \left(A_{n}\right) \equiv \infty$ , and $A=\varnothing$ .

Problem 10.4

The natural generalization of (4.9) is

\begin{align} \mu \left(\lim \inf _{n} A_{n}\right) & \leq \lim \inf _{n} \mu \left(A_{n}\right) \tag {10.6}\\ & \leq \lim \sup _{n} \mu \left(A_{n}\right) \leq \mu \left(\lim \sup _{n} A_{n}\right). \end{align}

Show that the left-hand inequality always holds. Show that the right-hand inequality holds if $\mu \left(U_{k \geq n} A_{k}\right)<\infty$ for some $n$ but can fail otherwise.

?

Problem 10.5

3.10 A measure space $(\Omega , \mathscr {F}, \mu )$ is complete if $A \subset B, B \in \mathscr {F}$ , and $\mu (B)=0$ together imply that $A \in \mathscr {F}$ -the definition is just as in the probability case. Use the ideas of Problem 3.10 to construct a complete measure space $\left(\Omega , \mathscr {F}^{+}, \mu^{+}\right)$ such that $\mathscr {F} \subset \mathscr {F}^{+}$ and $\mu$ and $\mu^{+}$ agree on $\mathscr {F}$ .

?

Problem 10.6

(a)(b)

The condition in Theorem 10.2(iv) essentially characterizes $\sigma$ -finiteness.

?

(a)

Suppose that $\left(\Omega , \mathscr {F}_{,} \mu \right)$ has no "infinite atoms," in the sense that for every $A$ in $\mathscr {F}$ , if $\mu (A)=\infty$ , then there is in $\mathscr {F}$ a $B$ such that $B \subset A$ and $0<\mu (B)<\infty$ . Show that if $\mathscr {F}$ does not contain an uncountable, disjoint collection of sets of positive measure, then $\mu$ is $\sigma$ -finite. (Use Zorn's lemma.)

(b)

Show by example that this is false without the condition that there are no "infinite atoms."

Problem 10.7

Example 10.5 shows that Theorem 10.3 fails without the $\sigma$ -finiteness condition. Construct other examples of this kind.

?