Measures

Prove that if $A$ and $B$ are subsets of $\mathbb {R}$ and $\left|B\right| = 0$, then $\left|A \cup B\right| = \left|A\right|$.

Suppose $A \subseteq \mathbb {R}$ and $t \in \mathbb {R}$. Let $t A = \left\{ t a: a \in A\right\}$. Prove that $\left|t A\right| = \left|t\right|\left|A\right|$.

Prove that if $A, B \subseteq \mathbb {R}$ and $\left|A\right| < \infty$, then $\left|B \backslash A\right| \geq \left|B\right|-\left|A\right|$.

Suppose $F$ is a subset of $\mathbb {R}$ with the property that every open cover of $F$ has a finite subcover. Prove that $F$ is closed and bounded.

Suppose $\mathcal{A}$ is a set of closed subsets of $\mathbb {R}$ such that $\bigcap_{F \in \mathcal{A}} F = \varnothing$. Prove that if $\mathcal{A}$ contains at least one bounded set, then there exist $n \in \mathbb {N}$ and $F_{1}, \ldots , F_{n} \in \mathcal{A}$ such that $F_{1} \cap \cdots \cap F_{n} = \varnothing$.

Prove that if $a, b \in \mathbb {R}$ and $a < b$, then

Suppose $a, b, c, d$ are real numbers with $a < b$ and $c < d$. Prove that

Prove that if $A \subseteq \mathbb {R}$ and $t > 0$, then $\left|A\right| = \left|A \cap (-t, t)\right|+\left|A \cap (\mathbb {R} \backslash (-t, t))\right|$.

Prove that $\left|A\right| = \lim_{t \rightarrow \infty }\left|A \cap (-t, t)\right|$ for all $A \subseteq \mathbb {R}$.

Prove that $\left|[0,1] \backslash \mathbf{Q}\right| = 1$.

Prove that if $I_{1}, I_{2}, \ldots$ is a disjoint sequence of open intervals, then

Suppose $r_{1}, r_{2}, \ldots$ is a sequence that contains every rational number. Let

Suppose $\varepsilon > 0$. Prove that there exists a subset $F$ of $[0,1]$ such that $F$ is closed, every element of $F$ is an irrational number, and $\left|F\right| > 1-\varepsilon$.

Consider the following figure, which is drawn accurately to scale.

Show that $\mathcal{S} = \left\{ \bigcup_{n \in K}(n, n+1]: K \subseteq \mathbf{Z}\right\}$ is a $\sigma$-algebra on $\mathbb {R}$.

Verify both bullet points in Example 2.28.

Suppose $\mathcal{S}$ is the smallest $\sigma$-algebra on $\mathbb {R}$ containing $\left\{ (r, s]: r, s \in \mathbf{Q}\right\}$. Prove that $\mathcal{S}$ is the collection of Borel subsets of $\mathbb {R}$.

Suppose $\mathcal{S}$ is the smallest $\sigma$-algebra on $\mathbb {R}$ containing $\left\{ (r, n]: r \in \mathbf{Q}, n \in \mathbf{Z}\right\}$. Prove that $\mathcal{S}$ is the collection of Borel subsets of $\mathbb {R}$.

Suppose $\mathcal{S}$ is the smallest $\sigma$-algebra on $\mathbb {R}$ containing $\left\{ (r, r+1): r \in \mathbf{Q}\right\}$. Prove that $\mathcal{S}$ is the collection of Borel subsets of $\mathbb {R}$.

Suppose $\mathcal{S}$ is the smallest $\sigma$-algebra on $\mathbb {R}$ containing $\left\{ [r, \infty ): r \in \mathbf{Q}\right\}$. Prove that $\mathcal{S}$ is the collection of Borel subsets of $\mathbb {R}$.

Prove that the collection of Borel subsets of $\mathbb {R}$ is translation invariant. More precisely, prove that if $B \subseteq \mathbb {R}$ is a Borel set and $t \in \mathbb {R}$, then $t+B$ is a Borel set.

Prove that the collection of Borel subsets of $\mathbb {R}$ is dilation invariant. More precisely, prove that if $B \subseteq \mathbb {R}$ is a Borel set and $t \in \mathbb {R}$, then $t B$ (which is defined to be $\left\{ t b: b \in B\right\}$) is a Borel set.

Give an example of a measurable space $(X, \mathcal{S})$ and a function $f: X \rightarrow \mathbb {R}$ such that $\left|f\right|$ is $\mathcal{S}$-measurable but $f$ is not $\mathcal{S}$-measurable.

Show that the set of real numbers that have a decimal expansion with the digit 5 appearing infinitely often is a Borel set.

Suppose $\mathcal{T}$ is a $\sigma$-algebra on a set $Y$ and $X \in \mathcal{T}$. Let $\mathcal{S} = \left\{ E \in \mathcal{T}: E \subseteq X\right\}$.

Suppose $f: \mathbb {R} \rightarrow \mathbb {R}$ is a function.

Suppose $(X, \mathcal{S})$ is a measurable space, $E_{1}, \ldots , E_{n}$ are disjoint subsets of $X$, and $c_{1}, \ldots , c_{n}$ are distinct nonzero real numbers. Prove that $c_{1} \chi_{E_{1}}+\cdots +c_{n} \chi_{E_{n}}$ is an $\mathcal{S}$-measurable function if and only if $E_{1}, \ldots , E_{n} \in \mathcal{S}$.

Suppose $X$ is a set and $E_{1}, E_{2}, \ldots$ is a disjoint sequence of subsets of $X$ such that $\bigcup_{k = 1}^{\infty } E_{k} = X$. Let $\mathcal{S} = \left\{ \bigcup_{k \in K} E_{k}: K \subseteq \mathbb {N}\right\}$.

Suppose $\mathcal{S}$ is a $\sigma$-algebra on a set $X$ and $A \subseteq X$. Let

Suppose $X$ is a Borel subset of $\mathbb {R}$ and $f: X \rightarrow \mathbb {R}$ is a function such that $\left\{ x \in X: f\text{ is not continuous at }x\right\}$ is a countable set. Prove $f$ is a Borel measurable function.

Suppose $f: \mathbb {R} \rightarrow \mathbb {R}$ is differentiable at every element of $\mathbb {R}$. Prove that $f^{\prime }$ is a Borel measurable function from $\mathbb {R}$ to $\mathbb {R}$.

Suppose $X$ is a nonempty set and $\mathcal{S}$ is the $\sigma$-algebra on $X$ consisting of all subsets of $X$ that are either countable or have a countable complement in $X$. Give a characterization of the $\mathcal{S}$-measurable real-valued functions on $X$.

Suppose $(X, \mathcal{S})$ is a measurable space and $f, g: X \rightarrow \mathbb {R}$ are $\mathcal{S}$-measurable functions. Prove that if $f(x) > 0$ for all $x \in X$, then $f^{g}$ (which is the function whose value at $x \in X$ equals $f(x)^{g(x)}$) is an $\mathcal{S}$-measurable function.

Prove 2.52.

Suppose $B \subseteq \mathbb {R}$ and $f: B \rightarrow \mathbb {R}$ is an increasing function. Prove that $f$ is continuous at every element of $B$ except for a countable subset of $B$.

Suppose $f: \mathbb {R} \rightarrow \mathbb {R}$ is a strictly increasing function. Prove that the inverse function $f^{-1}: f(\mathbb {R}) \rightarrow \mathbb {R}$ is a continuous function.

Suppose $f: \mathbb {R} \rightarrow \mathbb {R}$ is a strictly increasing function and $B \subseteq \mathbb {R}$ is a Borel set. Prove that $f(B)$ is a Borel set.

Suppose $B \subseteq \mathbb {R}$ and $f: B \rightarrow \mathbb {R}$ is an increasing function. Prove that there exists a sequence $f_{1}, f_{2}, \ldots$ of strictly increasing functions from $B$ to $\mathbb {R}$ such that

for every $x \in B$.

Suppose $B \subseteq \mathbb {R}$ and $f: B \rightarrow \mathbb {R}$ is a bounded increasing function. Prove that there exists an increasing function $g: \mathbb {R} \rightarrow \mathbb {R}$ such that $g(x) = f(x)$ for all $x \in B$.

Prove or give a counterexample: If $(X, \mathcal{S})$ is a measurable space and

is a function such that $f^{-1}((a, \infty )) \in \mathcal{S}$ for every $a \in \mathbb {R}$, then $f$ is an $\mathcal{S}$-measurable function.

Suppose $f: B \rightarrow \mathbb {R}$ is a Borel measurable function. Define $g: \mathbb {R} \rightarrow \mathbb {R}$ by

Prove that $g$ is a Borel measurable function.

Give an example of a measurable space $(X, \mathcal{S})$ and a family $\left\{ f_{t}\right\}_{t \in \mathbb {R}}$ such that each $f_{t}$ is an $\mathcal{S}$-measurable function from $X$ to $[0,1]$, but the function $f: X \rightarrow [0,1]$ defined by

is not $\mathcal{S}$-measurable.

Show that

for every $x \in \mathbb {R}$.

Explain why there does not exist a measure space $(X, \mathcal{S}, \mu )$ with the property that $\left\{ \mu (E): E \in \mathcal{S}\right\} = [0,1)$.

Let $2^{\mathbb {N}}$ denote the $\sigma$-algebra on $\mathbb {N}$ consisting of all subsets of $\mathbb {N}$. Suppose $\mu$ is a measure on $\left(\mathbb {N}, 2^{\mathbb {N}}\right)$. Prove that there is a sequence $w_{1}, w_{2}, \ldots$ in $[0, \infty ]$ such that

for every set $E \subseteq \mathbb {N}$.

Give an example of a measure $\mu$ on $\left(\mathbb {N}, 2^{\mathbb {N}}\right)$such that

Give an example of a measure space $(X, \mathcal{S}, \mu )$ such that

Suppose $(X, \mathcal{S}, \mu )$ is a measure space such that $\mu (X) < \infty$. Prove that if $\mathcal{A}$ is a set of disjoint sets in $\mathcal{S}$ such that $\mu (A) > 0$ for every $A \in \mathcal{A}$, then $\mathcal{A}$ is a countable set.

Find all $c \in [3, \infty )$ such that there exists a measure space $(X, \mathcal{S}, \mu )$ with

Give an example of a measure space $(X, \mathcal{S}, \mu )$ such that

Give an example of a set $X$, a $\sigma$-algebra $\mathcal{S}$ of subsets of $X$, a set $\mathcal{A}$ of subsets of $X$ such that the smallest $\sigma$-algebra on $X$ containing $\mathcal{A}$ is $\mathcal{S}$, and two measures $\mu$ and $\nu$ on $(X, \mathcal{S})$ such that $\mu (A) = \nu (A)$ for all $A \in \mathcal{A}$ and $\mu (X) = \nu (X) < \infty$, but $\mu \neq \nu$.

Suppose $\mu$ and $\nu$ are measures on a measurable space $(X, \mathcal{S})$. Prove that $\mu +\nu$ is a measure on $(X, \mathcal{S})$.

Give an example of a measure space $(X, \mathcal{S}, \mu )$ and a decreasing sequence $E_{1} \supseteq E_{2} \supseteq \cdots$ of sets in $\mathcal{S}$ such that

Suppose $(X, \mathcal{S}, \mu )$ is a measure space and $C, D, E \in \mathcal{S}$ are such that

Find and prove a formula for $\mu (C \cup D \cup E)$ in terms of $\mu (C), \mu (D), \mu (E)$, $\mu (C \cap D), \mu (C \cap E), \mu (D \cap E)$, and $\mu (C \cap D \cap E)$.

Suppose $X$ is a set and $\mathcal{S}$ is the $\sigma$-algebra of all subsets $E$ of $X$ such that $E$ is countable or $X \backslash E$ is countable. Give a complete description of the set of all measures on $(X, \mathcal{S})$.

Prove that there exists a bounded set $A \subseteq \mathbb {R}$ such that $\left|F\right| \leq \left|A\right|-1$ for every closed set $F \subseteq A$.

Prove that there exists a set $A \subseteq \mathbb {R}$ such that $\left|G \backslash A\right| = \infty$ for every open set $G$ that contains $A$.

The phrase nontrivial interval is used to denote an interval of $\mathbb {R}$ that contains more than one element. Recall that an interval might be open, closed, or neither.

Prove that if $A \subseteq \mathbb {R}$ is Lebesgue measurable, then there exists an increasing sequence $F_{1} \subseteq F_{2} \subseteq \cdots$ of closed sets contained in $A$ such that

Suppose $A \subseteq \mathbb {R}$ and $\left|A\right| < \infty$. Prove that $A$ is Lebesgue measurable if and only if for every $\varepsilon > 0$ there exists a set $G$ that is the union of finitely many disjoint bounded open intervals such that $\left|A \backslash G\right|+\left|G \backslash A\right| < \varepsilon$.

Prove that if $A \subseteq \mathbb {R}$ is Lebesgue measurable, then there exists a decreasing sequence $G_{1} \supseteq G_{2} \supseteq \cdots$ of open sets containing $A$ such that

Prove that the collection of Lebesgue measurable subsets of $\mathbb {R}$ is translation invariant. More precisely, prove that if $A \subseteq \mathbb {R}$ is Lebesgue measurable and $t \in \mathbb {R}$, then $t+A$ is Lebesgue measurable.

Prove that the collection of Lebesgue measurable subsets of $\mathbb {R}$ is dilation invariant. More precisely, prove that if $A \subseteq \mathbb {R}$ is Lebesgue measurable and $t \in \mathbb {R}$, then $t A$ (which is defined to be $\left\{ t a: a \in A\right\}$) is Lebesgue measurable.

Prove that if $A$ and $B$ are disjoint subsets of $\mathbb {R}$ and $B$ is Lebesgue measurable, then $\left|A \cup B\right| = \left|A\right|+\left|B\right|$.

Prove that if $A \subseteq \mathbb {R}$ and $\left|A\right| > 0$, then there exists a subset of $A$ that is not Lebesgue measurable.

Suppose $b < c$ and $A \subseteq (b, c)$. Prove that $A$ is Lebesgue measurable if and only if $\left|A\right|+\left|(b, c) \backslash A\right| = c-b$.

Suppose $A \subseteq \mathbb {R}$. Prove that $A$ is Lebesgue measurable if and only if

for every $n \in \mathbb {N}$.

Show that $\frac{1}{4}$ and $\frac{9}{13}$ are both in the Cantor set.

Show that $\frac{13}{17}$ is not in the Cantor set.

List the eight open intervals whose union is $G_{4}$ in the definition of the Cantor set (2.74).

Let $C$ denote the Cantor set. Prove that $\left\{ \frac{1}{2} x+\frac{1}{2} y: x, y \in C\right\} = [0,1]$.

Prove that every open interval of $\mathbb {R}$ contains either infinitely many or no elements in the Cantor set.

Evaluate $\int_{0}^{1} \Lambda$, where $\Lambda$ is the Cantor function.

Evaluate each of the following:

Find each of the following sets:

Show that there exists a function $f: \mathbb {R} \rightarrow \mathbb {R}$ such that the image under $f$ of every nonempty open interval is $\mathbb {R}$.

For $A \subseteq \mathbb {R}$, the quantity

is called the inner measure of $A$.

Suppose $X$ is a finite set. Explain why a sequence of functions from $X$ to $\mathbb {R}$ that converges pointwise on $X$ also converges uniformly on $X$.

Give an example of a sequence of functions from $\mathbb {N}$ to $\mathbb {R}$ that converges pointwise on $\mathbb {N}$ but does not converge uniformly on $\mathbb {N}$.

Give an example of a sequence of continuous functions $f_{1}, f_{2}, \ldots$ from $[0,1]$ to $\mathbb {R}$ that converges pointwise to a function $f:[0,1] \rightarrow \mathbb {R}$ that is not a bounded function.

Prove or give a counterexample: If $A \subseteq \mathbb {R}$ and $f_{1}, f_{2}, \ldots$ is a sequence of uniformly continuous functions from $A$ to $\mathbb {R}$ that converges uniformly to a function $f: A \rightarrow \mathbb {R}$, then $f$ is uniformly continuous on $A$.

Give an example to show that Egorov's Theorem can fail without the hypothesis that $\mu (X) < \infty$.

Suppose $(X, \mathcal{S}, \mu )$ is a measure space with $\mu (X) < \infty$. Suppose $f_{1}, f_{2}, \ldots$ is a sequence of $\mathcal{S}$-measurable functions from $X$ to $\mathbb {R}$ such that $\lim_{k \rightarrow \infty } f_{k}(x) = \infty$ for each $x \in X$. Prove that for every $\varepsilon > 0$, there exists a set $E \in \mathcal{S}$ such that $\mu (X \backslash E) < \varepsilon$ and $f_{1}, f_{2}, \ldots$ converges uniformly to $\infty$ on $E$ (meaning that for every $t > 0$, there exists $n \in \mathbb {N}$ such that $f_{k}(x) > t$ for all integers $k \geq n$ and all $x \in E$).

Suppose $F$ is a closed bounded subset of $\mathbb {R}$ and $g_{1}, g_{2}, \ldots$ is an increasing sequence of continuous real-valued functions on $F$ (thus $g_{1}(x) \leq g_{2}(x) \leq \cdots$ for all $x \in F$) such that $\sup \left\{ g_{1}(x), g_{2}(x), \ldots \right\} < \infty$ for each $x \in F$. Define a real-valued function $g$ on $F$ by

Prove that $g$ is continuous on $F$ if and only if $g_{1}, g_{2}, \ldots$ converges uniformly on $F$ to $g$.

Suppose $\mu$ is the measure on $\left(\mathbb {N}, 2^{\mathbb {N}}\right)$ defined by

Prove that for every $\varepsilon > 0$, there exists a set $E \subseteq \mathbb {N}$ with $\mu \left(\mathbb {N} \backslash E\right) < \varepsilon$ such that $f_{1}, f_{2}, \ldots$ converges uniformly on $E$ for every sequence of functions $f_{1}, f_{2}, \ldots$ from $\mathbb {N}$ to $\mathbb {R}$ that converges pointwise on $\mathbb {N}$.

Suppose $F_{1}, \ldots , F_{n}$ are disjoint closed subsets of $\mathbb {R}$. Prove that if

is a function such that $\left.g\right|_{F_{k}}$ is a continuous function for each $k \in \left\{ 1, \ldots , n\right\}$, then $g$ is a continuous function.

Suppose $F \subseteq \mathbb {R}$ is such that every continuous function from $F$ to $\mathbb {R}$ can be extended to a continuous function from $\mathbb {R}$ to $\mathbb {R}$. Prove that $F$ is a closed subset of $\mathbb {R}$.

Prove or give a counterexample: If $F \subseteq \mathbb {R}$ is such that every bounded continuous function from $F$ to $\mathbb {R}$ can be extended to a continuous function from $\mathbb {R}$ to $\mathbb {R}$, then $F$ is a closed subset of $\mathbb {R}$.

Give an example of a Borel measurable function $f$ from $\mathbb {R}$ to $\mathbb {R}$ such that there does not exist a set $B \subseteq \mathbb {R}$ such that $\left|\mathbb {R} \backslash B\right| = 0$ and $\left.f\right|_{B}$ is a continuous function on $B$.

Prove or give a counterexample: If $f_{t}: \mathbb {R} \rightarrow \mathbb {R}$ is a Borel measurable function for each $t \in \mathbb {R}$ and $f: \mathbb {R} \rightarrow (-\infty , \infty ]$ is defined by

then $f$ is a Borel measurable function.

Suppose $b_{1}, b_{2}, \ldots$ is a sequence of real numbers. Define $f: \mathbb {R} \rightarrow [0, \infty ]$ by