Abstract Measure

Prove that the counting measure introduced in Example 5.1.4 is a measure on $\mathbb {P}(X)$.

Example 5.1.4: Let $X$ be a set and $\mathfrak {M} = \mathbb {P}(X)$. For $E \subset X$ define $\mu (E) = n$ if $E$ has exactly $n$ elements and $\mu (E) = \infty$ if $E$ is an infinite set. Then $\mu$ is called the counting measure on $X$.

Prove that the Dirac measure introduced in Example 5.1.5 is a measure on $\mathbb {P}(X)$.

Example 5.1.5: Let $x_{0} \in X$ and let $\mu (E) = 1$ if $x_{0} \in E$ and $\mu (E) = 0$ if $x_{0} \notin E$. This is the Dirac measure $\delta_{x_{0}}$.

Prove the assertions made in Example 5.1.6.

Example 5.1.6: Let $(X, \mathfrak {M}, \mu )$ be a measure space and let $E \in \mathfrak {M}$. The function $\mu_{E}$ on $\mathfrak {M}$ defined by $\mu_{E}(A) = \mu \left(E \cap A\right)$ is a measure. Also, the set $\mathfrak {M}_{E} = \left\{ E \cap A : A \in \mathfrak {M}\right\}$ is a $\sigma$-algebra. Therefore, $(E, \mathfrak {M}_{E}, \mu_{E})$ is a measure space.

Let $\mu$ be a function defined on $\mathbb {P}(\mathbb {R})$ by

Prove or disprove: $\mu$ is a measure.

Let $X=\left\{ x_{k}\right\}_{k=1}^{\infty }$ and suppose that $\left\{ p_{k}\right\}_{k=1}^{\infty }$ is a sequence of nonnegative numbers such that $\sum_{n=1}^{\infty } p_{n}=1$. For any set $A \subset X$ we define $N_{A}=\left\{ k \in \mathbb {N} : x_{k} \in A\right\}$ and

Prove that $\mu$ is a measure on $\mathbb {P}(X)$.

Let $\left\{ \mu_{n}\right\}$ be a sequence of measures on a measurable space ( $X, \mathfrak {M}$ ) for which there is a constant $M \geq 0$ such that $\mu_{n}(X) \leq M$, for all $n \in \mathbb {N}$. Define

Prove that $\mu$ is a measure on $\mathfrak {M}$.

Prove Theorem 5.1.9 (ii). Give an example to show that the assumption $\mu \left(B_{1}\right)<\infty$ cannot be omitted.

Theorem 5.1.9 (ii) (Continuity of Measure): Let $(X, \mathfrak {M}, \mu )$ be a measure space. If $\left\{ B_{n}\right\}_{n=1}^{\infty }$ is a decreasing sequence in $\mathfrak {M}$ and if $\mu \left(B_{1}\right) < \infty$ then $\mu \left(\cap_{n=1}^{\infty } B_{n}\right) = \lim_{n \rightarrow \infty } \mu \left(B_{n}\right)$.

Let $(X, \mathfrak {M}, \mu )$ be a measure space. Prove that $\mu$ is subadditive on $\mathfrak {M}$.

Let $\left\{ \mu_{n}\right\}_{n=1}^{\infty }$ be a sequence of measures on a $\sigma$-algebra $\mathcal{A}$, and let $\left\{ \mu_{n}\right\}$ converge to a function $\mu$. In other words, suppose that for every $A \in \mathcal{A}$, $\lim \mu_{n}(A)=\mu \left(A\right)$. Also, suppose that there is a finite measure $\nu$ on $\mathcal{A}$, such that for every $n \in \mathbb {N}$ and for every set $A \in \mathcal{A}, \mu_{n}(A) \leq \nu \left(A\right)$. Prove that $\mu$ is a measure of $\mathcal{A}$. Does this remain true if there is no such measure $\nu$ ?

Let $(X, \mathfrak {M})$ be a measurable space and let $\left\{ E_{n}\right\} \subset \mathfrak {M}$. Prove that the set $A$ of all those elements of $X$ that belong to infinitely many sets from the sequence $\left\{ E_{n}\right\}$ is measurable.

Let $\left\{ \mu_{n}\right\}$ be a monotone sequence of measures on a measurable space ( $X, \mathfrak {M}$ ) and let $\left\{ \mu_{n}\right\}$ converge to a function $\mu$. Prove or disprove: $\mu$ is a measure on $\mathfrak {M}$.

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space. Let $\mathfrak {M}^{\prime }$ be the collection of all $E \subset X$ for which there exist $A, B \in \mathfrak {M}$ such that $A \subset E \subset B$ and $\mu \left(B \backslash A\right)=0$. For such $E$ define $\mu \left(E\right)=\mu \left(A\right)$. Prove that $\mathfrak {M}^{\prime }$ is a $\sigma$-algebra and that $\mu$ is a measure on $\mathfrak {M}^{\prime }$.

Let ( $X, \mathfrak {M}, \mu$ ) be a $\sigma$-finite measure space and let $E \in \mathfrak {M}$ for which $\mu \left(E\right) \neq 0$. Prove that there exists a measurable set $A \subset E$ for which $0< \mu \left(A\right)<\infty$.

Define $E=\left\{ x \in [0,1]:\left|x-\frac{p}{q}\right|<q^{-3} \text{ for infinitely many } p, q \in \mathbb {N}\right\}$. Prove that $m\left(E\right)=0$.

Let $(X, \mathfrak {M}, \mu )$ be a measure space and suppose that there is $n \in \mathbb {N}$ and a set of real numbers $A=\left\{ a_{1}, a_{2}, \ldots , a_{n}\right\}$ such that for all $E \in \mathfrak {M}, \mu \left(E\right) \in A$.

Let $(X, \mathfrak {M}, \mu )$ be a $\sigma$-finite measure space and let $\mathcal{A}$ be a disjoint family of subsets of $X$. Prove that there can be only countably many sets in $\mathcal{A}$ that have positive measure.

Let $\mathfrak {M}$ be a $\sigma$-algebra of subsets of $X$ and let $f: X \rightarrow Y$. Prove or disprove: If $\mu$ is a measure on $X$, then $\lambda \left(E\right):=\mu \left(f^{-1}(E)\right)$ defines a measure on $Y$.

Let ( $X, \mathfrak {M}$ ) be a measurable space and let $\mu$ be a finitely additive measure. Suppose that $\mu$ is $\sigma$-subadditive. Prove that $\mu$ is a measure.

Let ( $X, \mathfrak {M}$ ) be a measurable space and let $\mu$ be a finitely additive measure. Suppose that $\mu$ has the following property: If $\left\{ A_{n}\right\}$ is a decreasing sequence of sets in $\mathfrak {M}$ such that $\cap_{n} A_{n}=\emptyset$, then $\lim_{n} \mu \left(A_{n}\right)=0$. Prove that $\mu$ is a measure.

Let $(X, \mathfrak {M}, \mu )$ be a finite measure space and let $\nu$ be a finitely additive measure. Suppose that $\nu$ has the following property: If $\left\{ A_{n}\right\} \subset \mathfrak {M}$ such that $\lim_{n} \mu \left(A_{n}\right)=0$, then $\lim_{n} \nu \left(A_{n}\right)=0$. Prove that $\nu$ is a measure.

Let $(X, \mathfrak {M})$ be a measurable space and let $\mu$ be a finitely additive measure. Suppose that $\mu$ has the following property: If $\left\{ A_{n}\right\}$ is an increasing sequence of sets in $\mathfrak {M}$, then $\mu \left(\bigcup_{n=1}^{\infty } A_{n}\right)=\lim_{n \rightarrow \infty } \mu \left(A_{n}\right)$. Prove that $\mu$ is a measure.

Let $(X, \mathfrak {M}, \mu )$ be a finite measure space and let $\left\{ E_{n}\right\} \subset \mathfrak {M}$. Prove that

Let $(X, \mathfrak {M}, \mu )$ be a finite measure space and let $\left\{ E_{n}\right\} \subset \mathfrak {M}$. Prove that

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space and let $A, B$ be sets in $\mathfrak {M}$ such that $\mu \left(A \triangle B\right)=0$. Prove that for any $E \in \mathfrak {M}, \mu \left(E \cap A\right)=\mu \left(E \cap B\right)$.

Let $\mu$ and $\nu$ be measured on a measurable space ( $X, \mathfrak {M}$ ), and define $\lambda \left(E\right)=\min \left\{ \mu \left(E\right), \nu \left(E\right)\right\}$ and $\omega (E)=\max \left\{ \mu \left(E\right), \nu \left(E\right)\right\}$, for $E \in \mathfrak {M}$. Prove or disprove:

Let $\mu$ and $\nu$ be measured on a measurable space ( $X, \mathfrak {M}$ ), and let $\alpha \geq 0$. Prove that:

Let ( $X, \mathfrak {M}, \mu$ ) be a probability space, i.e., a measure space such that $\mu \left(X\right)=1$, and let $\left\{ A_{n}\right\} \subset \mathfrak {M}$ be independent, meaning that for any $n \in \mathbb {N}$ and any positive integers $1 \leq k_{1}<\cdots <k_{n}$,

Prove that the sequence $\left\{ A_{n}^{C}\right\}$ is also independent.

Let $(X, \mathfrak {M}, \mu )$ be a probability space, i.e., a measure space such that $\mu \left(X\right)=1$, and let $\left\{ A_{n}\right\} \subset \mathfrak {M}$. Prove that $\mu \left(\cap_{k=1}^{n} A_{k}\right) \geq \sum_{k=1}^{n} \mu \left(A_{k}\right)-(n-1)$.

Let $\mu$ and $\nu$ be measures on a measurable space ( $X, \mathfrak {M}$ ) and suppose that $\mu \geq \nu$. Prove that:

Let $X$ be a nonempty set, let $x_{0} \in X$, and consider the measure space $\left(X, \mathbb {P}(X), \delta_{x_{0}}\right)$. Prove that two functions $f, g: X \rightarrow \overline{\mathbb {R}}$ are equal a.e. if and only if $f\left(x_{0}\right)=g\left(x_{0}\right)$.

Let $X$ be a nonempty set, let $\mu$ be the counting measure on $X$, and consider the measure space $(X, \mathbb {P}(X), \mu )$. Prove that two functions $f, g: X \rightarrow \overline{\mathbb {R}}$ are equal a.e. if and only if $f(x)=g(x)$ for every $x \in X$.

Prove that $\mu^{*}$ defined in Example 5.2.5 is an outer measure.

Example 5.2.5: Let $\mathfrak {E}$ be a ring on $X$ and for $A \subset X$ let $\mu^{*}(A) := \inf \left\{ \sum_{k=1}^{\infty } \mu (E_{k}) : \left\{ E_{k} : k \in \mathbb {N}\right\} \subset \mathfrak {E} \text{ and } A \subset \cup_{k=1}^{\infty } E_{k}\right\}$.

Let $(X, \mathfrak {M})$ be a measurable space and let $\left\{ \mu_{n}^{*}\right\}$ be a sequence of outer measures on $(X, \mathfrak {M})$. Prove that $\sum_{n} \mu_{n}^{*}$ and $\sup_{n} \mu_{n}^{*}$ are also outer measures.

Let $\mathcal{A}$ be a collection of subsets of $X$ that includes the empty set, let $F: \mathcal{A} \rightarrow [0,+\infty ]$ such that $F(\emptyset )=0$, and let

Prove that $\mu^{*}$ is an outer measure.

In Exercises 5.2.4-5.2.7, $X$ is a set and $\mu^{*}: \mathbb {P}(X) \rightarrow [0,+\infty ]$ is given. Prove that $\mu^{*}$ is an outer measure and determine the $\sigma$-algebra $\mathfrak {M}$ of all measurable sets.

$\mu^{*}(A)$ is the number of points in $A$ if $A$ is a finite set, and $\mu^{*}(A)=\infty$ when $A$ is an infinite set.

In Exercises 5.2.4-5.2.7, $X$ is a set and $\mu^{*}: \mathbb {P}(X) \rightarrow [0,+\infty ]$ is given. Prove that $\mu^{*}$ is an outer measure and determine the $\sigma$-algebra $\mathfrak {M}$ of all measurable sets.

$\mu^{*}(\varnothing )=0$ and $\mu^{*}(E)=1$ for $E \neq \varnothing$.

In Exercises 5.2.4-5.2.7, $X$ is a set and $\mu^{*}: \mathbb {P}(X) \rightarrow [0,+\infty ]$ is given. Prove that $\mu^{*}$ is an outer measure and determine the $\sigma$-algebra $\mathfrak {M}$ of all measurable sets.

$\mu^{*}(\varnothing )=0,\mu^{*}(X)=2$, and $\mu^{*}(E)=1$ for all other subsets of $X$.

In Exercises 5.2.4-5.2.7, $X$ is a set and $\mu^{*}: \mathbb {P}(X) \rightarrow [0,+\infty ]$ is given. Prove that $\mu^{*}$ is an outer measure and determine the $\sigma$-algebra $\mathfrak {M}$ of all measurable sets.

$\mu^{*}(A)=0$ if $A$ is a countable set and $\mu^{*}(A)=1$ if $A$ is uncountable.

Let $X$ be an infinite set, let $\nu$ be the counting measure on $X$, and let $F: \mathbb {P}(X) \rightarrow [0,+\infty ]$ be defined by $F(A)=\sqrt{\nu \left(A\right)}$.

Let $\mu$ be a function defined on subsets of $\mathbb {R}$ by setting $\mu \left(A\right)$ to be the number of integers in $A$.

Let $X$ be a nonempty set and let $\mu$ be a function defined on $\mathfrak {E}=\left\{ \varnothing , X\right\}$ by $\mu \left(\varnothing \right)=0$ and $\mu \left(X\right)=1$. Determine the outer measure $\mu^{*}$ induced by $\mu$ and the $\sigma$-algebra of measurable sets.

Let $X$ be a set with at least 2 points, let $A$ be a proper, nonempty subset of $X$, and let $\mu$ be a function defined on $\mathfrak {E}=\left\{ A, X\right\}$ by $\mu \left(A\right)=1$ and $\mu \left(X\right)=2$. Determine the outer measure $\mu^{*}$ induced by $\mu$ and the $\sigma$-algebra of measurable sets.

Let $\mathfrak {E}=\left\{ \varnothing ,[0,1],[0,3],[2,3]\right\} \subset \mathbb {P}(\mathbb {R})$, and let $\mu$ be a function defined on $\mathfrak {E}$ by $\mu \left(\emptyset \right)=0, \mu \left([0,1]\right)=\mu \left([0,3]\right)=\mu \left([2,3]\right)=1$. Determine the outer measure $\mu^{*}$ induced by $\mu$ and the $\sigma$-algebra of measurable sets.

Prove Proposition 5.2.14. Does the result remain true if $n$ is replaced by $\infty$ ?

Proposition 5.2.14: If $E_{1}, E_{2}, \ldots , E_{n}$ are pairwise disjoint sets in $\mathfrak {M}$ and $A$ is any subset of $X$ then $\mu^{*}\left(A \cap \left(\cup_{i=1}^{n} E_{i}\right)\right) = \sum_{i=1}^{n} \mu^{*}(A \cap E_{i})$. In particular, $\mu^{*}\left(\cup_{i=1}^{n} E_{i}\right) = \sum_{i=1}^{n} \mu^{*}(E_{i})$.

Let $\mu^{*}$ be an outer measure on a set $X$, let $\left\{ A_{n}\right\}$ be a sequence of subsets of $X$, and let $\left\{ E_{n}\right\}$ be a sequence of pairwise disjoint, measurable subsets of $X$ such that $A_{n} \subset E_{n}$, for all $n \in \mathbb {N}$. Prove that

Let $\mu^{*}$ be an outer measure and suppose that it is additive. Prove that it is countably additive, hence a measure.

Let $\mu^{*}$ be an outer measure on a set $X$ and let $A \subset X$ have outer measure 0. Prove that

for any set $B \subset X$.

Let $\mu^{*}$ be an outer measure on a set $X$ and let $A \subset X$. Prove that $A$ is measurable if and only if for each $\varepsilon >0$ there exists a measurable set $E$ such that $E \subset A$ and $\mu^{*}(A \backslash E)<\varepsilon$.

Let $\mu^{*}$ be an outer measure on a set $X$ and let $A \subset X$. Suppose that for each $\varepsilon >0$ there exists a measurable set $E$ such that $\mu^{*}(A \triangle E)<\varepsilon$. Prove that $A$ is measurable.

Let $\mu^{*}$ be an outer measure on a set $X$, let $A \subset X$, and let $E$ be a measurable subset of $X$. Prove that

Let $\left(X, \mathfrak {M}_{1}, \mu_{1}\right)$ and $\left(X, \mathfrak {M}_{2}, \mu_{2}\right)$ be measure spaces. Prove that $\mu_{1}$ and $\mu_{2}$ generate the same outer measure on $X$ (in the sense of Example 5.2.5) if and only if both $\mu_{1}=\mu_{2}^{*}$ on $\mathfrak {M}_{1}$ and $\mu_{2}=\mu_{1}^{*}$ on $\mathfrak {M}_{2}$.

Example 5.2.5: Let $\mathfrak {E}$ be a ring on $X$, $\mu$ a measure on $\mathfrak {E}$, and for $A \subset X$ let $\mu^{*}(A) := \inf \left\{ \sum_{k=1}^{\infty } \mu (E_{k}) : \left\{ E_{k} : k \in \mathbb {N}\right\} \subset \mathfrak {E} \text{ and } A \subset \cup_{k=1}^{\infty } E_{k}\right\}$. Then $\mu^{*}$ is an outer measure, and $\mu^{*}(A)=\mu (A)$ if $A \in \mathfrak {E}$.

Let $(X, \mathfrak {M}, \mu )$ be a measure space and $\mu^{*}$ the outer measure on $X$ as defined in Example 5.2.5. Prove that if $E \subset X$ then there exists a measurable set $B$ that contains $E$ and $\mu \left(B\right)=\mu^{*}(E)$.

Example 5.2.5: Let $\mathfrak {E}$ be a ring on $X$, $\mu$ a measure on $\mathfrak {E}$, and for $A \subset X$ let $\mu^{*}(A) := \inf \left\{ \sum_{k=1}^{\infty } \mu (E_{k}) : \left\{ E_{k} : k \in \mathbb {N}\right\} \subset \mathfrak {E} \text{ and } A \subset \cup_{k=1}^{\infty } E_{k}\right\}$. Then $\mu^{*}$ is an outer measure, and $\mu^{*}(A)=\mu (A)$ if $A \in \mathfrak {E}$.

Let $\mathfrak {E}$ be a ring on $X$, let $\mu$ be a measure on $\mathfrak {E}$, and let $\mu^{*}$ be the outer measure on $X$ as in Example 5.2.5. We will denote by $\mathfrak {E}_{\sigma }$ the collection of all countable unions of sets in $\mathfrak {E}$ and by $\mathfrak {E}_{\sigma \delta }$ the collection of all countable intersections of sets in $\mathfrak {E}_{\sigma }$. Prove that:

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space such that $\mu \left(X\right)=1$ and $\mu^{*}$ an outer measure on $X$. Suppose that $E \subset X$ and $\mu^{*}(E)=1$. Prove that if $A, B \in \mathfrak {M}$ and $A \cap E=B \cap E$ then $\mu \left(A\right)=\mu \left(B\right)$.

Prove that if $\mu^{*}(E)=0$ then $E \in \mathfrak {M}$ and $\mu \left(E\right)=0$.

Let $h:[a, b] \rightarrow \mathbb {R}$ be a bounded, left-continuous, increasing function. Prove that there is a $\sigma$-algebra $\mathfrak {M} \subset \mathbb {P}([a, b])$ and a measure $\mu$ on $\mathfrak {M}$ such that $\mu \left([x, y\right))=h(y)-h(x)$, for all $x<y$ in $[a, b]$. Remark 5.2.19. This measure is known as the Lebesgue-Stieltjes measure associated to $h$.

Give an example of a collection $\mathcal{A}$ of subsets of $X$ and measures $\mu , \nu$ on $\sigma (\mathcal{A})$ such that $\mu \left(A\right)=\nu \left(A\right)$, for all $A \in \mathcal{A}$, but there is a set $E \in \sigma (\mathcal{A})$ for which $\mu \left(E\right) \neq \nu \left(E\right)$.

Let $E \subset X$ be such that $E \cap A \in \mathfrak {M}$ whenever $A \in \mathfrak {M}$ and $\mu \left(A\right)<\infty$. Prove that $E \in \mathfrak {M}$.

Prove that the Carathéodory extension of a measure $\mu^{\prime }$ is unique if $\mu^{\prime }$ is $\sigma$-finite on $\mathfrak {E}$.

Let $\mathfrak {E}$ be a ring that includes $X$ and suppose that $\mu$ is a finite measure on $\mathfrak {E}$.

Let $\left\{ r_{n}\right\}$ be an arrangement of $\mathbb {Q}$ in a sequence, let $\alpha_{n}(E)=1$ if $r_{n} \in E$ and $\alpha_{n}(E)=0$ otherwise, and let $\mu$ be a function defined on subsets of $\mathbb {R}$ by setting $\mu \left(E\right)=\sum_{n=1}^{\infty } 2^{-n} \alpha_{n}(E)$. Is $\mu$ additive? Is it $\sigma$-additive?

For $E \subset \mathbb {N}$ let $\alpha_{k}(E)=1$ if $k \in E$ and $\alpha_{k}(E)=0$ otherwise. Let $\mathfrak {M}$ be the collection of all sets for which $\mu \left(E\right)$ exists, where

Let $\mathcal{A}$ be the collection of finite unions of sets of the form $(a, b] \cap \mathbb {Q}$, where $-\infty \leq a \leq b \leq +\infty$, and define $\mu \left(\varnothing \right)=0$ and $\mu \left(A\right)=\infty$ for $A \neq \varnothing$. Prove that:

Let $X$ be a nonempty set, let $\mathcal{A} \subset \mathbb {P}(X)$, and let $\mu : \mathcal{A} \rightarrow [0,+\infty ]$. Prove or disprove: if $\mathfrak {M}$ is the $\sigma$-algebra obtained by Carathéodory's construction then $\mathcal{A} \subset \mathfrak {M}$.

Let $X$ be a nonempty set, let $E \subset X$, let $\mu^{*}$ be an outer measure on $\mathbb {P}(X)$, let $\mu$ be the restriction of $\mu^{*}$ to the $\sigma$-algebra $\mathfrak {M}$ of $\mu^{*}$-measurable sets, and let $\mu^{+}$be the outer measure associated to $\mu$. Prove that:

Let $(X, \mathfrak {M})$ be a measurable space and $\mu^{*}$ an outer measure on $X$. Prove that $\left|\mu^{*}(A)-\mu^{*}(B)\right| \leq \mu^{*}(A \triangle B)$ for any $A, B \subset X$, provided that at least one of the numbers $\mu^{*}(A), \mu^{*}(B)$ is finite.

Let $\mathfrak {E}$ be an algebra on $X$, let $\mu$ be a finite measure on $\mathfrak {E}$, let $\bar{\mu }$ be its Carathéodory extension on a $\sigma$-algebra $\mathfrak {M}$, let $E \in \mathfrak {M}$, and let $\varepsilon >0$. Prove that there is a set $A \in \mathfrak {E}$ such that $\bar{\mu }(A \triangle E)<\varepsilon$.

Prove that every algebra is a ring. Give an example to show that the converse is not true.

Prove that every ring that contains $X$ is also an algebra.

Suppose that $\mathcal{R}$ is a ring. Prove that the set

is an algebra.

Suppose that $\mathcal{R}$ is a ring. Prove that the set

is an algebra.

Let $\left\{ A_{n}\right\} \subset \mathbb {R}^{2}$ be an increasing sequence. Prove that $m^{*}\left(\cup_{n} A_{n}\right)= \lim m^{*}\left(A_{n}\right)$.

Prove that there is no Lebesgue measurable set $A \subset \mathbb {R}$ with the property that for any interval $I$,

Let $E$ be a Lebesgue measurable set such that $0<m\left(E\right)<\infty$ and let $\left\{ x_{n}\right\}$ be a sequence in $[0,1]$. Prove that there exists $y \in \mathbb {R}$ and a subsequence $\left\{ x_{n_{k}}\right\}$ such that

Show that if $A \subset [0,1]$ and $m\left(A\right)>0$, then there are $x$ and $y$ in $A$ such that $\left|x-y\right|$ is an irrational number.