Measurable Sets and Measure Spaces

Define $\mu : 2^{\mathbb {N}} \rightarrow [0, \infty )$ by $\mu (E) = \sum_{n \in E} 1 / 2^{n}$. Show that this is a finite measure.

For each $k$, let $\left(X, \mathcal{M}, \mu_{k}\right)$ be a measure space and suppose that there is a $c \geq 0$ for which $\mu_{k}(X) \leq c$ for all $k$. Show that $\mu = \sum_{k = 1}^{\infty } \frac{\mu_{k}}{2^{k}}$ also is a measure.

Let $\mathcal{M}$ be a $\sigma$-algebra and the set-function $\mu : \mathcal{M} \rightarrow [0, \infty )$ be finitely additive. Prove that $\mu$ is a measure if and only if whenever $\left\{ A_{k}\right\}_{k = 1}^{\infty }$ is an ascending collection of sets in $\mathcal{M}$, then (1) holds.

Let ($X, \mathcal{M}, \mu$) be a finite measure space. The symmetric difference, $A \Delta B$, of two measurable sets $A$ and $B$ is defined by

Identify two measurable sets whose symmetric difference has measure zero, and prove the following triangle inequality on $\mathcal{M}$ : For measurable sets $A, B$ and $C$,

The $\sigma$-algebra $\mathcal{M}$, with the metric $\rho (A, B) = \mu (A \Delta B)$, is called the Nikodym metric space.

Let $\mathcal{M}$ be the collection of subsets of $X$ that are either countable or have a countable complement in $X$. Show that $\mathcal{M}$ is a $\sigma$-algebra. For $E \in \mathcal{M}$, define $\mu (E) = 0$ if $E$ is countable and $\mu (E) = 1$, if $E$ has a countable complement. Show that $\mu$ is a measure. Is this measure space complete?

Let $\mathcal{S}$ be any collection of subsets of $X$ and define $\mathcal{M}$ to be the intersection of all $\sigma$-algebras of subsets of $X$ that contain $\mathcal{S}$. Show that $\mathcal{M}$ is a $\sigma$-algebra, and it is the smallest $\sigma$-algebra that contains $\mathcal{S}$, in the sense that it is contained in any $\sigma$-algebra that contains $\mathcal{S}$. For $X = \mathbb {R}$, find the smallest $\sigma$-algebra that contains all singleton sets.

Let $\mathcal{M}$ be a $\sigma$-algebra. Verify the following: