Measures and Their Properties

Explain why there does not exist a measure space $(X, \mathcal{S}, \mu )$ with the property that $\left\{ \mu (E): E \in \mathcal{S}\right\} = [0,1)$.

Let $2^{\mathbb {N}}$ denote the $\sigma$-algebra on $\mathbb {N}$ consisting of all subsets of $\mathbb {N}$. Suppose $\mu$ is a measure on $\left(\mathbb {N}, 2^{\mathbb {N}}\right)$. Prove that there is a sequence $w_{1}, w_{2}, \ldots$ in $[0, \infty ]$ such that

for every set $E \subseteq \mathbb {N}$.

Give an example of a measure $\mu$ on $\left(\mathbb {N}, 2^{\mathbb {N}}\right)$such that

Give an example of a measure space $(X, \mathcal{S}, \mu )$ such that

Suppose $(X, \mathcal{S}, \mu )$ is a measure space such that $\mu (X) < \infty$. Prove that if $\mathcal{A}$ is a set of disjoint sets in $\mathcal{S}$ such that $\mu (A) > 0$ for every $A \in \mathcal{A}$, then $\mathcal{A}$ is a countable set.

Find all $c \in [3, \infty )$ such that there exists a measure space $(X, \mathcal{S}, \mu )$ with

Give an example of a measure space $(X, \mathcal{S}, \mu )$ such that

Give an example of a set $X$, a $\sigma$-algebra $\mathcal{S}$ of subsets of $X$, a set $\mathcal{A}$ of subsets of $X$ such that the smallest $\sigma$-algebra on $X$ containing $\mathcal{A}$ is $\mathcal{S}$, and two measures $\mu$ and $\nu$ on $(X, \mathcal{S})$ such that $\mu (A) = \nu (A)$ for all $A \in \mathcal{A}$ and $\mu (X) = \nu (X) < \infty$, but $\mu \neq \nu$.

Suppose $\mu$ and $\nu$ are measures on a measurable space $(X, \mathcal{S})$. Prove that $\mu +\nu$ is a measure on $(X, \mathcal{S})$.

Give an example of a measure space $(X, \mathcal{S}, \mu )$ and a decreasing sequence $E_{1} \supseteq E_{2} \supseteq \cdots$ of sets in $\mathcal{S}$ such that

Suppose $(X, \mathcal{S}, \mu )$ is a measure space and $C, D, E \in \mathcal{S}$ are such that

Find and prove a formula for $\mu (C \cup D \cup E)$ in terms of $\mu (C), \mu (D), \mu (E)$, $\mu (C \cap D), \mu (C \cap E), \mu (D \cap E)$, and $\mu (C \cap D \cap E)$.

Suppose $X$ is a set and $\mathcal{S}$ is the $\sigma$-algebra of all subsets $E$ of $X$ such that $E$ is countable or $X \backslash E$ is countable. Give a complete description of the set of all measures on $(X, \mathcal{S})$.