Differentiation

Show that $\left|f(x)\right| \leq (M f)(x)$ at every Lebesgue point of $f$ if $f \in L^{1}\left(\mathbb {R}^{k}\right)$.

For $\delta > 0$, let $I(\delta )$ be the segment $(-\delta , \delta ) \subset \mathbb {R}^{1}$. Given $\alpha$ and $\beta , 0 \leq \alpha \leq \beta \leq 1$, construct a measurable set $E \subset \mathbb {R}^{1}$ so that the upper and lower limits of

are $\beta$ and $\alpha$, respectively, as $\delta \rightarrow 0$. (Compare this with Section 7.12).

Suppose that $E$ is a measurable set of real numbers with arbitrarily small periods. Explicitly, this means that there are positive numbers $p_{i}$, converging to 0 as $i \rightarrow \infty$, so that

Prove that then either $E$ or its complement has measure 0. Hint: Pick $\alpha \in \mathbb {R}^{1}$, put $F(x) = m(E \cap [\alpha , x])$ for $x > \alpha$, show that

if $\alpha +p_{i} < x < y$. What does this imply about $F^{\prime }(x)$ if $m(E) > 0$?

Call $t$ a period of the function $f$ on $\mathbb {R}^{1}$ if $f(x+t) = f(x)$ for all $x \in \mathbb {R}^{1}$. Suppose $f$ is a real Lebesgue measurable function with periods $s$ and $t$ whose quotient $s / t$ is irrational. Prove that there is a constant $c$ such that $f(x) = c$ a.e., but that $f$ need not be constant.

Hint: Apply Exercise 3 to the sets $\left\{ f > \lambda \right\}$.

If $A \subset \mathbb {R}^{1}$ and $B \subset \mathbb {R}^{1}$, define $A+B = \left\{ a+b: a \in A, b \in B\right\}$. Suppose $m(A) > 0, m(B) > 0$. Prove that $A+B$ contains a segment, by completing the following outline.

There are points $a_{0}$ and $b_{0}$ where $A$ and $B$ have metric density 1. Choose a small $\delta > 0$. Put $c_{0} = a_{0}+b_{0}$. For each $\epsilon$, positive or negative, define $B_{\epsilon }$ to be the set of all $c_{0}+\epsilon -b$ for which $b \in B$ and $\left|b-b_{0}\right| < \delta$. Then $B_{\epsilon } \subset \left(a_{0}+\epsilon -\delta , a_{0}+\epsilon +\delta \right)$. If $\delta$ was well chosen and $\left|\epsilon \right|$ is sufficiently small, it follows that $A$ intersects $B_{\epsilon }$, so that $A+B \supset \left(c_{0}-\epsilon_{0}, c_{0}+\epsilon_{0}\right)$ for some $\epsilon_{0} > 0$.

Let $C$ be Cantor's "middle thirds" set and show that $C+C$ is an interval, although $m(C) = 0$. (See also Exercise 19, Chap. 9).

Suppose $G$ is a subgroup of $\mathbb {R}^{1}$ (relative to addition), $G \neq \mathbb {R}^{1}$, and $G$ is Lebesgue measurable. Prove that then $m(G) = 0$.

Hint: Use Exercise 5.

Construct a continuous monotonic function $f$ on $\mathbb {R}^{1}$ so that $f$ is not constant on any segment although $f^{\prime }(x) = 0$ a.e.

Let $V = (a, b)$ be a bounded segment in $\mathbb {R}^{1}$. Choose segments $W_{n} \subset V$ in such a way that their union $W$ is dense in $V$ and the set $K = V-W$ has $m(K) > 0$. Choose continuous functions $\varphi_{n}$ so that $\varphi_{n}(x) = 0$ outside $W_{n}, 0 < \varphi_{n}(x) < 2^{-n}$ in $W_{n}. \operatorname {Put} \varphi = \sum \varphi_{n}$ and define

Prove the following statements:

Suppose $0 < \alpha < 1$. Pick $t$ so that $t^{\alpha } = 2$. Then $t > 2$, and the construction of Example (b) in Sec. 7.16 can be carried out with $\delta_{n} = (2 / t)^{n}$. Show that the resulting function $f$ belongs to Lip $\alpha$ on $[0,1]$.

If $f \in \operatorname {Lip} 1$ on $[a, b]$, prove that $f$ is absolutely continuous and that $f^{\prime } \in L^{\infty }$.

Assume that $1 < p < \infty , f$ is absolutely continuous on $[a, b], f^{\prime } \in L^{p}$, and $\alpha = 1 / q$, where $q$ is the exponent conjugate to $p$. Prove that $f \in \operatorname {Lip} \alpha$.

Suppose $\varphi :[a, b] \rightarrow \mathbb {R}^{1}$ is nondecreasing.

Let $B V$ be the class of all $f$ on $[a, b]$ that have bounded variation on $[a, b]$, as defined after Theorem 7.19. Prove the following statements.

Show that the product of two absolutely continuous functions on $[a, b]$ is absolutely continuous. Use this to derive a theorem about integration by parts.

Construct a monotonic function $f$ on $\mathbb {R}^{1}$ so that $f^{\prime }(x)$ exists (finitely) for every $x \in \mathbb {R}^{1}$, but $f^{\prime }$ is not a continuous function.

Suppose $E \subset [a, b], m(E) = 0$. Construct an absolutely continuous monotonic function $f$ on $[a, b]$ so that $f^{\prime }(x) = \infty$ at every $x \in E$.

Hint $: E \subset \bigcap V_{n}, V_{n}$ open, $m\left(V_{n}\right) \subset 2^{-n}$. Consider the sum of the characteristic functions of these sets.

Suppose $\left\{ \mu_{n}\right\}$ is a sequence of positive Borel measures on $\mathbb {R}^{k}$ and

Assume $\mu \left(\mathbb {R}^{k}\right) < \infty$. Show that $\mu$ is a Borel measure. What is the relation between the Lebesgue decompositions of the $\mu_{n}$ and that of $\mu$?

Prove that

Derive corresponding theorems for sequences $\left\{ f_{n}\right\}$ of positive nondecreasing functions on $\mathbb {R}^{1}$ and their sums $f = \sum f_{n}$.

Let $\varphi_{0}(t) = 1$ on $[0,1), \varphi_{0}(t) = -1$ on [1, 2), extend $\varphi_{0}$ to $\mathbb {R}^{1}$ so as to have period 2, and define $\varphi_{n}(t) = \varphi_{0}\left(2^{n} t\right), n = 1,2,3, \ldots$

Assume that $\sum \left|c_{n}\right|^{2} < \infty$ and prove that the series

converges then for almost every $t$. Probabilistic interpretation: The series $\sum \left(\pm c_{n}\right)$ converges with probability 1. Suggestion: $\left\{ \varphi_{n}\right\}$ is orthonormal on $[0,1]$, hence $\left(^{*}\right)$ is the Fourier series of some $f \in L^{2}$. If $a = j \cdot 2^{-N}, b = (j+1) \cdot 2^{-N}, a < t < b$, and $s_{N} = c_{1} \varphi_{1}+\cdots +c_{N} \varphi_{N}$, then, for $n > N$,

and the last integral converges to $\int_{a}^{b} f d m$, as $n \rightarrow \infty$. Show that $\left(^{*}\right)$ converges to $f(t)$ at almost every Lebesgue point of $f$.

Suppose $f$ is continuous on $\mathbb {R}^{1}, f(x) > 0$ if $0 < x < 1, f(x) = 0$ otherwise. Define

Prove that

If $f$ is a real function on $[0,1]$ and

the length of the graph of $f$ is, by definition, the total variation of $\gamma$ on $[0,1]$. Show that this length is finite if and only if $f \in B V$. (See Exercise 13). Show that it is equal to

if $f$ is absolutely continuous.

The definition of Lebesgue points, as made in Sec. 7.6, applies to individual integrable functions, not to the equivalence classes discussed in Sec. 3.10. However, if $F \in L^{1}\left(\mathbb {R}^{k}\right)$ is one of these equivalence classes, one may call a point $x \in \mathbb {R}^{k}$ a Lebesgue point of $F$ if there is a complex number, let us call it $(S F)(x)$, such that

for one (hence for every) $f \in F$. Define $(S F)(x)$ to be 0 at those points $x \in \mathbb {R}^{k}$ that are not Lebesgue points of $F$. Prove the following statement: If $f \in F$, and $x$ is a Lebesgue point of $f$, then $x$ is also a Lebesgue point of $F$, and $f(x) = (S F)(x)$. Hence $S F \in F$.

Thus $S$ "selects" a member of $F$ that has a maximal set of Lebesgue points.