Measure and Integration

Write out the details of the proof, using diagonalization, that $\mathcal{B}$ is uncountable.

Prove that $\mathcal{B}_{T}$ is countable, and use this result to give an alternative proof (as outlined in the text) that $\mathcal{B}$ is uncountable.

In this exercise you are asked to determine the set $B_{E}$ for given events $E$.

Prove that $(0,1] \backslash S$ is uncountable, where $S$ is the set referred to in the strong law of large numbers. Hint: Consider the map from $(0,1]$ to itself that maps the binary expression $\omega =. a_{1} a_{2} \ldots$ to.$a_{1} 11 a_{2} 11 a_{3} 11 a_{4} \ldots$. Prove that this map is one-to-one and its image is contained in $(0,1] \backslash S$.

Let $\mathcal{R}$ be a $\sigma$-ring. Prove that $\bigcap_{n=1}^{\infty } A_{n} \in \mathcal{R}$ whenever $A_{n} \in \mathcal{R}, n= 1,2, \ldots$. Hint: Verify, and use, that $\bigcap_{n=1}^{\infty } A_{n}=A_{1} \backslash \bigcup_{n=1}^{\infty }\left(A_{1} \backslash A_{n}\right)$.

Assume that $\mu$ is a nonnegative, additive function defined on a ring $\mathcal{R}$.

Assume that $\mu$ is a countably additive function defined on a ring $\mathcal{R}$, that $A_{n} \in \mathcal{R}, A \in \mathcal{R}$, that

and that

Prove that

(Hint: Put $B_{1}=A_{1}$, and $B_{n}=A_{n} \backslash A_{n-1}$ for $n=2,3, \ldots$ )

Prove that $\mathcal{E}$ is a ring, but is not a $\sigma$-ring.

Prove Lemma 3.1. (Hint: Consider the case that $A$ is an interval first, and then consider finite unions of disjoint intervals.)

Lemma 3.1: If $A \in \mathcal{E}$ and $\varepsilon > 0$, then there exists a closed set $F \in \mathcal{E}$ and an open set $G \in \mathcal{E}$ such that $F \subseteq A \subseteq G$ and $m(F) \geq m(A) - \varepsilon$ and $m(G) \leq m(A) + \varepsilon$.

Prove that $m^{*}$ is countably subadditive.

This exercise is about the symmetric difference and distance functions $S$ and $D$ defined on $2^{X}$.

If you have studied some abstract algebra, you may know a different use of the term "ring" (the definition is given, incidentally, at the beginning of Section 6.6). In this exercise, the term "ring" refers to the algebraic notion. Prove that $2^{\left(R^{n}\right)}$ becomes a commutative ring with "multiplication" of two sets taken to be their intersection, and with "addition" of two sets taken to be their symmetric difference.

Prove that the Cantor set is in $\mathcal{M}$ and that it has Lebesgue measure zero.

Prove that every continuous function is measurable.

Give an example of a function $f$ such that $f$ is not measurable but $\left|f\right|$ is measurable.

Which characteristic functions are (Lebesgue) integrable on $\mathbb {R}$? Is the characteristic function of the rational numbers integrable on the unit interval? If so, what is value of this integral?

Prove Theorem 3.8. (Hint: One approach is to notice that

)

Theorem 3.8: If $f$ is measurable, then $\left|f\right|$ is measurable.

Fill in the details of the proof of Theorem 3.9.

Theorem 3.9: If $f$ and $g$ are measurable, then so are $f+g$, $fg$, $f^{+}$, and $f^{-}$. If $\left\{ f_{k}\right\}_{k=1}^{\infty }$ is a sequence of measurable functions, then the four functions

are also measurable.

Complete the proof of Theorem 3.10.

Theorem 3.10: If $f$ is a real-valued function defined on $\mathbb {R}^{n}$, then there exists a sequence $\left\{ s_{k}\right\}_{k=1}^{\infty }$ of simple functions such that $\lim_{k \to \infty } s_{k}(x) = f(x)$ for every $x \in \mathbb {R}^{n}$. Further, if $f$ is measurable, then the $s_{k}$'s may be chosen to be measurable simple functions. Finally, if $f \geq 0$, then $\left\{ s_{k}\right\}_{k=1}^{\infty }$ may be chosen to satisfy $s_{1} \leq s_{2} \leq \cdots$.

Prove that $f \in \mathcal{L}\left(\mathbb {R}^{n}\right)$ if and only if $\int_{\mathbb {R}^{n}}\left|f\right| d m<\infty$.

Supply proofs for the missing parts of Theorem 3.11.

Theorem 3.11: The Lebesgue integral enjoys several properties. (a) The integral is linear: $\int_{E} cf\, dm = c\int_{E} f\, dm$ and $\int_{E} (f+g)\, dm = \int_{E} f\, dm + \int_{E} g\, dm$ whenever $f, g \in \mathcal{L}(\mathbb {R}^{n})$, $c \in \mathbb {R}$, and $E \in \mathcal{M}$. (b) The integral is monotone: $\int_{E} f\, dm \leq \int_{E} g\, dm$ whenever $f, g \in \mathcal{L}(\mathbb {R}^{n})$, $f(x) \leq g(x)$ for all $x \in E$. (c) For every $f \in \mathcal{L}(\mathbb {R}^{n})$, we have $\left|f\right| \in \mathcal{L}(\mathbb {R}^{n})$ and $\left|\int_{E} f\, dm\right| \leq \int_{E} \left|f\right|\, dm$. (d) For every $f \in \mathcal{L}(\mathbb {R}^{n})$, we have $\int_{E} f\, dm = 0$ for every measurable set $E$ of measure zero; from this it follows that $\int_{A} f\, dm = \int_{B} f\, dm$ whenever $A$ and $B$ are measurable sets, $B \subseteq A$, and $m(A \setminus B) = 0$.

Supply proofs for the missing parts:

You have read about the phrase "almost everywhere" in the text. In particular, we say that two measurable functions are "equal almost everywhere" if the set of points where they differ has measure zero.

Give an example to show that strict inequality can hold in Fatou's lemma.

Give an example to show that without the existence of the function $g$ in the dominated convergence theorem, the conclusion may fail.

Prove that the triples $(X, \mathcal{R}, \mu )$ given in Examples 4 and 6 at the beginning of Section 6 are, in fact, measure spaces. For the measure in Example 3, show that

whenever $n \neq m$ implies $\left[a_{n}, b_{n}\right] \cap \left[a_{m}, b_{m}\right]=\emptyset$.

For $0<p<1$, we can define the $L^{p}$-space and $\left\| \cdot \right\|_{p}$ in the same way that we did for $1 \leq p<\infty$. Prove, by giving a suitable example, that $\left\| \cdot \right\|_{p}$ does not satisfy the triangle inequality, and hence is not a norm.

Prove, as outlined in the text, that

for $x, y \geq 0$ and Hölder conjugates $p$ and $q$ with $1<p<\infty$.

Prove that

for $f, g \in L^{p}(\mu )$ and $1<p<\infty$. (This equality is used in the proof of Theorem 3.19.)

Prove that $L^{\infty }$, with norm $\left\| \cdot \right\|_{\infty }$, is a normed linear space.

In this exercise you will investigate relations between the various $L^{p}$-spaces.

Show that for each $f \in L^{\infty },\left|f(x)\right| \leq \left\| f\right\|_{\infty }$ almost everywhere.

Let $I$ be an interval in $\mathbb {R}$.

The point of this exercise is to show that the norm on $L^{p}$, for $p \neq 2$, does not come from an inner product. We start by considering $L^{p}([-1,1])$, with Lebesgue measure. Set $f(x)=1+x$ and $g(x)=1-x$.