General Measure Spaces: Their Properties and Construction

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:

Show that in the construction of Lebesgue measure on $\mathbb {R}$, the outer-measure is unchanged if covers by open, bounded intervals are replaced by covers by general, bounded intervals.

Let $\mu^{*}: 2^{X} \rightarrow [0, \infty ]$ be an outer-measure. Let $A \subseteq X,\left\{ E_{k}\right\}_{k = 1}^{\infty }$ be a countable, disjoint collection of measurable sets and $E = \bigcup_{k = 1}^{\infty } E_{k}$. Show that

Show that any measure that is induced by an outer-measure is complete.

Define $\mu : 2^{\mathbb {R}} \rightarrow \mathbb {R}$ by defining $\mu (A)$ to be the number of integers in $A$. Determine the outer-measure $\mu^{*}$ induced by $\mu$ and the $\sigma$-algebra of measurable sets.

Let $S \subseteq 2^{X}$ and $\mu^{*}: 2^{X} \rightarrow [0, \infty ]$ be the outer-measure induced by $\mu : S \rightarrow [0, \infty ]$. Show that for a set $A \subseteq X$ to be measurable with respect to $\mu^{*}$ it suffices to verify that

not for all of the subsets $E$ of $X$. This was the definition of measurable set used by Lebesgue for the set-function length defined on the collection of open, bounded intervals in $\mathbb {R}$.

Show that a non-negative set-function on a $\sigma$-algebra is a measure if and only if it is a premeasure.

Let $\mathbb {Q}$ be the set of rational numbers and $\mathcal{S}$ the collection of all finite unions of intervals of the form $(a, b] \cap \mathbb {Q}$, where $a, b \in \mathbb {Q}$ and $a \leq b$. Define $\mu (a, b] = \infty$ if $a < b$ and $\mu (\emptyset ) = 0$. Show that $\mathcal{S}$ is closed with respect to relative complements and $\mu : \mathcal{S} \rightarrow [0, \infty ]$ is a premeasure. Then show that the extension of $\mu$ to the smallest $\sigma$-algebra containing $\mathcal{S}$ is not unique.

Verify the assertions made in this section's first and second example.

Show that the three collections of sets in this section's third example are semi-rings.

Let $\mathcal{S}$ be a $\sigma$-algebra of subsets of $X$ and $\mu : \mathcal{S} \rightarrow [0, \infty ]$ be a measure. Let $\bar{\mu }: \overline{\mathcal{M}} \rightarrow [0, \infty ]$ be the Carathéodory measure induced by $\mu$. Show that $\mathcal{S}$ is a subcollection of $\overline{\mathcal{M}}$ and it may be a proper subcollection.

Show that Lebesgue on $\mathbb {R}$ is the Carathéodory extension of the premeasure Lebesgue measure on the semi-ring of Borel sets.