Beyond Lebesgue $\sigma$-Algebra

Prove that the relation $\sim$ introduced in the proof of Theorem 5.4.2 is an equivalence relation.

In the proof of Theorem 5.4.2 (which shows that any $E \subset \mathbb {R}$ with $m(E) > 0$ contains a non-measurable subset), the relation $\sim$ is defined on $E$ by $x \sim y$ if and only if $x - y \in \mathbb {Q}$.

Prove Theorem 5.4.3.

Theorem 5.4.3: There are disjoint sets $A, B \subset \mathbb {R}$ such that $m^{*}(A \cup B) < m^{*}(A) + m^{*}(B)$.

Show that there is a continuous, strictly increasing function on the interval $[0,1]$ that maps a set of positive measures onto a set of measure zero.

Let $f$ be a continuous function on a set $E \subset \mathbb {R}$, and let $A$ be a Lebesgue measurable subset of $\mathbb {R}$. Prove or disprove: $f^{-1}(A)$ is measurable.

Let $A$ be a nonmeasurable set constructed in the proof of Theorem 5.4.2 and let $B \subset A$ be measurable. Prove that $m\left(B\right)=0$.

Theorem 5.4.2: If $E \subset \mathbb {R}$ and $m(E)>0$ then there exists a set $A \subset E$ that is not measurable. In its proof, $E$ is assumed bounded, $E \subset [-b, b]$; on $E$ one defines $x \sim y$ if and only if $x-y \in \mathbb {Q}$, an equivalence relation, and, by the Axiom of Choice, $C_{E}$ is a set containing exactly one representative of each equivalence class. With $\Lambda =[-2b, 2b] \cap \mathbb {Q}$, the sets $\left\{ C_{E}+\lambda : \lambda \in \Lambda \right\}$ are pairwise disjoint and $E \subset \cup_{\lambda \in \Lambda }\left(C_{E}+\lambda \right)$, which shows that $A=C_{E}$ is not measurable.

Let $C_{E}$ be as in Theorem 5.4.2.

Let $A \subset \mathbb {R}$ have the property that

Prove or disprove: $A$ is measurable.

Show that the Borel $\sigma$-algebra $\mathcal{B}$ is generated by any of the following collections of subsets of $\mathbb {R}^{2}$ :

1. The closed sets.

2. The compact sets.

3. The open disks.

4. The rectangles.

5. The elementary sets.

Let $f: \mathbb {R} \rightarrow \mathbb {R}$ be a continuous increasing function and let $E$ be a Borel set. Prove that $f(E)$ is also a Borel set.

Prove that if a $\sigma$-algebra of subsets of $\mathbb {R}$ contains all intervals of the form ( $a,+\infty$ ), then it contains all intervals.

Show that every interval is a Borel set.

Let $X=\mathbb {R}^{2}$, let $\mathfrak {M}=\mathcal{B}_{2}$, the $\sigma$-algebra of Borel subsets of $\mathbb {R}^{2}$, and let $\mu =m_{2}$, the two-dimensional Lebesgue measure on $\mathcal{B}_{2}$. Let $D=\left\{ (x, x): x \in \mathbb {R}\right\}$. Prove that $D \in \mathcal{B}_{2}$ and that $m_{2}\left(D\right)=0$.

Prove that the $\sigma$-algebra $\mathcal{M}$ obtained in Exercise 5.2.25 contains all Borel subsets of $[a, b]$.

Let $\mu$ be a finite Borel measure on $\mathbb {R}^{n}$, and for $x \in \mathbb {R}^{n}$ let $f(x)= \mu \left(B(x, 1)\right)$, where $B(x, 1)$ denotes the open ball centered at $x$ of radius 1 . Prove that $f$ attains its minimum on each compact set in $\mathbb {R}^{n}$.

Let $E \in \mathcal{B}_{n}, F \in \mathcal{B}_{m}$. Prove that $E \times F \in \mathcal{B}_{n+m}$.

Prove that every measurable set in $\mathbb {R}^{n}$ is a union of a Borel set and a set of measure 0 .

Let $\mathcal{A}$ be a countably generated $\sigma$-algebra and let $\mathcal{C} \subset \mathcal{A}$ be a $\sigma$ algebra. Proves or disprove: $\mathcal{C}$ is countably generated.

Prove or disprove: there is a $\sigma$-finite measure on $\mathcal{B}\left(\mathbb {R}^{n}\right)$ such that every Borel set with a nonempty interior has infinite measure.

Let $\mathcal{F}$ be a family of open intervals such that their endpoints form a dense subset of $\mathbb {R}$. Prove or disprove: $\sigma (\mathcal{F})=\mathcal{B}$.

Prove that the subset of $\mathbb {P}(\mathbb {R})$ consisting of the sets of the form $A \triangle B$, where $A \in \mathcal{B}$ and $B \in \mathcal{M}$ is a set of measure zero, is a $\sigma$-algebra.

Let $\mu$ be a finite Borel measure on $\mathbb {R}$ and let $A$ be a closed subset of $\mathbb {R}$. Prove that $f: \mathbb {R} \rightarrow \mathbb {R}$ defined by $f(x)=\mu \left(A+x\right)$ is upper semicontinuous.

Suppose that $X$ is a metric space. Prove that every Borel measurable subset of $X$ can be approximated from the inside by a closed set, i.e., if $E \subset X$ is a Borel set of finite measure and $\varepsilon >0$ then there exists a closed set $F \subset E$ such that $\mu \left(E \backslash F\right)<\varepsilon$.

Let $E \subset \mathbb {R}^{n}$ be a nonmeasurable set of finite outer measures. Show that there is a $G_{\delta }$ set $G$ such that

Prove Theorem 5.4.13.

Theorem 5.4.13: Let $(X, \mathfrak {M}, \mu )$ be a measure space. Define $\mathfrak {M}_{0}$ to be the collection of subsets $E$ of $X$ of the form $E = A \cup B$, where $B \in \mathfrak {M}$ and $A$ is contained in a set of $\mu$-measure $0$. For such a set $E$ define $\mu_{0}(E) = \mu (B)$. Then $\mathfrak {M}_{0}$ is a $\sigma$-algebra that contains $\mathfrak {M}$, $\mu_{0}$ is a unique measure that extends $\mu$ to $\mathfrak {M}_{0}$, and $(X, \mathfrak {M}_{0}, \mu_{0})$ is complete. Further, $\mathfrak {M}_{0}$ is the smallest such $\sigma$-algebra.

Prove that any measure obtained from Carathéodory's extension theorem is complete. In particular, the Lebesgue measure is complete.

Let $(X, \mathfrak {M}, \mu )$ be a complete measure space. Prove that if $A \cup B \in \mathfrak {M}$ and $\mu \left(B\right)=0$ then $A \in \mathfrak {M}$.

Let ( $X, \mathfrak {M}, \mu$ ) be a complete measure space. Prove that if $A, B \subset X$ are such that $A \in \mathfrak {M}$ and $\mu \left(A \triangle B\right)=0$ then $B \in \mathfrak {M}$ and $\mu \left(B\right)=\mu \left(A\right)$.

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space and let ( $X, \mathfrak {M}_{0}, \mu_{0}$ ) be its completion. Prove that $\mu^{*}(E)=\mu_{0}(E)$ for all $E \in \mathfrak {M}_{0}$.

Prove Theorem 5.4.15.

Theorem 5.4.15: The Lebesgue measure space $(\mathbb {R}^{n}, \mathcal{M}_{n}, m_{n})$ is the completion of the Borel measure space $(\mathbb {R}^{n}, \mathcal{B}_{n}, m_{n}|_{\mathcal{B}_{n}})$. (The completion of a measure space $(X, \mathfrak {M}, \mu )$ is the measure space $(X, \mathfrak {M}_{0}, \mu_{0})$ constructed in Theorem 5.4.13: $\mathfrak {M}_{0}$ consists of all sets $E = A \cup B$ with $B \in \mathfrak {M}$ and $A$ contained in a set of $\mu$-measure $0$, and $\mu_{0}(E) = \mu (B)$.)

Let $(X, \mathfrak {M}, \mu )$ be a measure space. Define $\mathfrak {M}_{1}$ to be the collection of subsets $E$ of $X$ for which there exist sets $A, B \in \mathfrak {M}$ such that $A \subset E \subset B$ and $\mu \left(B \backslash A\right)=0$. For such a set $E$ define $\mu_{1}(E)=\mu \left(A\right)$. Prove that $\left(X, \mathfrak {M}_{1}, \mu_{1}\right)$ is the (minimal) completion of ( $X, \mathfrak {M}, \mu$ ).