The $\sigma$-Algebra of Lebesgue Measurable Sets

Let $A, B \subset \mathbb {R}^{2}$, let $A \in \mathcal{M}$, and suppose that $m^{*}(A \triangle B)=0$. Prove that $B \in \mathcal{M}$ and $m\left(B\right)=m\left(A\right)$.

Prove that a set $B \subset \mathbb {R}^{2}$ is nonmeasurable if and only if there exists $\varepsilon >0$ such that for every measurable set $A \subset B, m^{*}(B \backslash A) \geq \varepsilon$.

Prove Corollary 4.3.9.

Corollary 4.3.9: Let $E \subset \mathbb {R}^{2}$. There exists a $G_{\delta }$ set $G \subset \mathbb {R}^{2}$ such that $E \subset G$ and $m(G) = m^{*}(E)$.

Let $A, B \subset \mathbb {R}^{2}$ be sets of finite outer measure such that $A \cup B$ is measurable and $m\left(A \cup B\right)=m^{*}(A)+m^{*}(B)$. Prove that $A, B$ are both measurable sets.

Prove that the Lebesgue outer measure on $\mathbb {R}^{2}$ is translation invariant.

Let $A \subset \mathbb {R}^{2}$ have a finite outer measure. Prove that $A^{C}-A^{C}=\mathbb {R}^{2}$.

The purpose of this exercise is to establish the following result: If $A \subset \mathbb {R}^{2}$ is a Lebesgue measurable set of positive measures, then the set $A-A:=\left\{ x-y: x \in A, y \in A\right\}$ contains a neighborhood of the origin.

Let $\mathfrak {M}$ be a $\sigma$-algebra of subsets of $X$ and let $f: X \rightarrow Y$. Prove or disprove: $f(\mathfrak {M})$ is a $\sigma$-algebra.

Let $\mathfrak {M}$ be a $\sigma$-algebra of subsets of $X$ and let $f: X \rightarrow Y$. Prove or disprove: $\mathfrak {N}:=\left\{ E: f^{-1}(E) \in \mathfrak {M}\right\}$ is a $\sigma$-algebra.

Let $f: X \rightarrow Y$ and let $\mathfrak {N}$ be a $\sigma$-algebra of subsets of $Y$. Prove or disprove: $\mathfrak {M}:=\left\{ f^{-1}(E): E \in \mathfrak {N}\right\}$ is a $\sigma$-algebra.

Let $X=\cup_{n=1}^{\infty } X_{n}$ and let $\mathfrak {M}_{n}$ be a $\sigma$-algebra on $X_{n}$, for each $n \in \mathbb {N}$. Prove that the set

is a $\sigma$-algebra on $X$.

Let $X$ be an uncountable set and let

Prove that $\mathcal{A}$ is a $\sigma$-algebra.

Let $\mathcal{A}_{1}, \mathcal{A}_{2}$ be $\sigma$-algebras on $X$. Prove or disprove: $\mathcal{A}_{1} \cap \mathcal{A}_{2}$ is a $\sigma$ algebra.

Let $\mathcal{A}_{1}, \mathcal{A}_{2}$ be $\sigma$-algebras on $X$. Prove or disprove: $\mathcal{A}_{1} \cup \mathcal{A}_{2}$ is a $\sigma$ algebra.

Let $\left\{ \mathcal{A}_{n}\right\}_{n=1}^{\infty }$ be an increasing sequence of $\sigma$-algebras. Prove or disprove: $\cup_{n=1}^{\infty } \mathcal{A}_{n}$ is a $\sigma$-algebra.

Let $X \subset \mathbb {R}^{2}$ be a countable set and let

Prove that $\mathcal{A}$ is the smallest $\sigma$-algebra that contains $\left\{ x\right\}$ for each $x \in X$.

Let $\mathcal{S}$ be the collection of distinct subsets of $\mathbb {R}$ that are symmetric about the origin. (A set $A$ is symmetric if $x \in A$ if and only if $-x \in A$.) Prove that $\mathcal{S}$ is a $\sigma$-algebra.

If $\mathcal{S}$ is a collection of subsets of $\mathbb {R}^{2}$, we denote by $\sigma (\mathcal{S})$ the smallest $\sigma$-algebra that contains $\mathcal{S}$.

Let $X$ be a set and $\mathcal{A}$ a subset of $\mathbb {P}(X)$. Prove that there is a unique smallest $\sigma$-algebra of subsets of $X$ that contains $\mathcal{A}$.

Let $X$ be a set and $\mathcal{A}$ a $\sigma$-algebra of subsets of $X$. Suppose that $T: X \rightarrow X$ is a bijection such that $T(A) \in \mathcal{A}$ for all $A \in \mathcal{A}$. Prove that $\left\{ A \in \mathcal{A}: T(A)=A\right\}$ is a $\sigma$-algebra.

Let $X, Y$ be nonempty sets and let $\mathcal{A} \subset \mathbb {P}(X)$. Prove that $\sigma (\mathcal{A}) \times Y$ is a $\sigma$-algebra on $X \times Y$.

Let $X$ be a set and $\mathcal{A}$ a $\sigma$-algebra of subsets of $X$. Suppose that $S \subset X$ is a set that is not in $\mathcal{A}$. Prove that

Let $X$ be a set and $\mathcal{A}$ a nonempty subset of $\mathbb {P}(X)$. Prove that

Let $X$ be a set and $\mathcal{A}$ a countable collection of disjoint subsets of $X$ such that their union covers $X$. Describe $\sigma (\mathcal{A})$.

Let $X \subset \mathbb {R}^{2}$, let $\mathcal{S}$ be a collection of subsets of $X$, and let $A \in \sigma (\mathcal{S})$. Prove that there is a countable subset $\mathcal{C}$ of $\mathcal{S}$ such that $A \in \sigma (\mathcal{C})$.

In Exercises 4.3.26-4.3.28, $Y$ is a nonempty subset of $X$ and, for each $\mathcal{A} \subset \mathbb {P}(X), \mathcal{A}_{Y}=\left\{ A \cap Y: A \in \mathcal{A}\right\}$.

Prove that if $\mathcal{A}$ is a $\sigma$-algebra then $\mathcal{A}_{Y}$ is a $\sigma$-algebra on $Y$.

In Exercises 4.3.26-4.3.28, $Y$ is a nonempty subset of $X$ and, for each $\mathcal{A} \subset \mathbb {P}(X), \mathcal{A}_{Y}=\left\{ A \cap Y: A \in \mathcal{A}\right\}$.

Prove that $\left\{ E \subset X: E \cap Y \in \sigma \left(\mathcal{A}_{Y}\right)\right\}$ is a $\sigma$-algebra.

In Exercises 4.3.26-4.3.28, $Y$ is a nonempty subset of $X$ and, for each $\mathcal{A} \subset \mathbb {P}(X), \mathcal{A}_{Y}=\left\{ A \cap Y: A \in \mathcal{A}\right\}$.

Prove that $\sigma \left(\mathcal{A}_{Y}\right)=\sigma (\mathcal{A})_{Y}$.

Prove that a finitely generated $\sigma$-algebra is a finite set.

Prove or disprove: if $\mathcal{A}=\sigma \left(\left\{ A_{n}\right\}_{n \in \mathbb {N}}\right)$ then $\mathcal{A}$ is uncountable.