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Integral and Measure: From Rather Simple to Rather Complex

Vigirdas Mackevičius

Chapter 12

Families of Sets - all with Video Answers

Educators

Chapter Questions

11:11

Problem 1

$\mathcal{D}$ consists of a single set $D \subset E$.

Paul A.
Paul A.
California State Polytechnic University, Pomona
00:30

Problem 2

Fix a set $D \subset E . \mathcal{D}$ consists of all subsets of $E$ containing $D$, that is $\mathcal{D}=\{F: D \subset F \subset E\}$.

Amy Jiang
Amy Jiang
Numerade Educator
02:46

Problem 3

$\mathcal{D}$ is the family of all sets consisting of:
i) a single element;
ii) two elements.

James Kiss
James Kiss
Numerade Educator

Problem 4

Let $f: E \rightarrow E$ be a bijection (one-to-one map). A set $A \subset E$ is called $f$-invariant if $f(A) \subset A$ and $f^{-1}(A) \subset A . \mathcal{D}$ is the family set of all $f$-invariant sets.

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02:01

Problem 5

$E:=\mathbb{R}^3$. A set $A \subset E$ us called a cylinder if from $(x, y, z) \in A$ it follows that $\left\{\left(x, y, z^{\prime}\right): z^{\prime} \in \mathbb{R}\right\} \subset A . \mathcal{D}$ is the family of all cylinders.

Jeffrey Utley
Jeffrey Utley
Numerade Educator

Problem 6

$E:=\mathbb{R}^2, \mathcal{D}$ is the family of all subsets of $E$ that can be covered by a finite number of horizontal straight lines.

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02:03

Problem 7

Let $f: E_1 \rightarrow E_2$, and let $\mathcal{A}$ be an algebra ( $\sigma$-algebra) of subsets of $E_2$. Show that

$$
f^{-1}(\mathcal{A})=\left\{A \subset E_1: A=f^{-1}(B), B \in \mathcal{A}\right\}
$$

is an algebra ( $\sigma$-algebra) of subsets of $E_1$.

James Chok
James Chok
Numerade Educator
13:09

Problem 8

Recall that the Borel $\sigma$-algebra $\mathcal{B}=\mathcal{B}(\mathbb{R})$ on $\mathbb{R}$ is the $\sigma$-algebra generated by the family of open intervals.
i) Show that $\mathcal{B}$ is generated by the intervals $(-\infty, r)$ with rational $r$.
ii) Show that all intervals $I \subset \mathbb{R}$ are Borel sets, that is, belong to $\mathcal{B}$.
iii) Show that the set of irrational numbers is a Borel set.

Mengchun Cai
Mengchun Cai
Numerade Educator

Problem 9

Prove that, in $E=\mathbb{R}^k$, all open and all closed sets are Borel sets. Is this true for an arbitrary metric space $E$ ?

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Problem 10

Give an example of a set $E$ and a family $\mathcal{M}$ of its subsets that contains $\varnothing$ and $E$ and is a monotone family but not a $\sigma$-algebra.

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