Derivatives on the Line*

Extend Examples 31.1 and 31.3: Let $p_{0}, \ldots , p_{r-1}$ be nonnegative numbers adding to 1, where $r \geq 2$; suppose there is no $i$ such that $p_{i}=1$. Let $X_{1}$, $X_{2}, \ldots$ be independent, identically distributed random variables such that $\mathbb {P}\left(X_{n}=i\right)=p_{i}, 0 \leq i<r$, and put $X=\sum_{n=1}^{\infty } X_{n} r^{-n}$. Let $F$ be the distribution function of $X$. Show that $F$ is continuous. Show that $F$ is strictly increasing over the unit interval if and only if all the $p_{i}$ are strictly positive. Show that $F(x) \equiv x$ for $0 \leq x \leq 1$ if $p_{i} \equiv r^{-1}$ and that otherwise $F$ is singular; prove singularity by extending the arguments both of Example 31.1 and of Exampie 31.3. What is the analogue of (31.17)?

$\uparrow$ In Problem 31.1 take $r=3$ and $p_{0}=p_{2}=\frac{1}{2}, p_{1}=0$. The corresponding $F$ is called the Cantor function. The complement in $[0,1]$ of the Cantor set (see Problems 1.5 and 3.16) consists of the middle third $\left(\frac{1}{3}, \frac{2}{3}\right)$, the middle thirds $\left(\frac{1}{9}, \frac{2}{9}\right)$ and $\left(\frac{7}{9}, \frac{8}{9}\right)$, and so on. Show that $F$ is $\frac{1}{2}$ on the first of these intervals, $\frac{1}{4}$ on the second, $\frac{3}{4}$ on the third, and so on. Show by direct argument that $F^{\prime }=0$ except on a set of Lebesgue measure 0.

A real function $f$ of a real variable is a Lebesgue function if $[x: f(x) \leq \alpha ]$ is a Lebesgue set for each $\alpha$.

$\uparrow$ An arbitrary function $f$ on ( 0,1$]$ can be represented as a composition of a Lebesgue function $f_{1}$ and a Borel function $f_{2}$. For $x$ in ( 0,1 ], let $d_{n}(x)$ be the $n$th digit in its nonterminating dyadic expansion, and define $f_{2}(x)= \sum_{n=1}^{\infty } 2 d_{n}(x) / 3^{n}$. Show that $f_{2}$ is increasing and that $f_{2}(0,1]$ is contained in the Cantor set. Take $f_{1}(x)$ to be $f\left(f_{2}^{-1}(x)\right)$ if $x \in f_{2}(0,1]$ and 0 if $x \in (0,1]-f_{2}(0,1]$. Now show that $f=f_{1} f_{2}$.

Let $r_{1}, r_{2}, \ldots$ be an enumeration of the rationals in ( 0,1 ) and put $F(x)= \Sigma_{k r_{k} \leq x} 2^{-k}$. Define $\varphi$ by (14.5) and prove that it is continuous and singular.

Suppose that $\mu$ and $F$ are related by (31.18). If $F$ is not absolutely continuous, then $\mu (A)>0$ for some set $A$ of Lebesgue measure 0. It is an interesting fact, however, that almost all translates of $A$ must have $\mu$-measure 0. From Fubini's theorem and the fact that $\lambda$ is invariant under translation and reflection through 0, show that, if $\lambda (A)=0$ and $\mu$ is $\sigma$-finite, then $\mu (A+x)=0$ for $x$ outside a set of Lebesgue measure 0.

$17.431 .6 \uparrow$ Show that $F$ is absolutely continuous if and only if for each Borel set $A, \mu (A+x)$ is continuous in $x$.

Let $F_{*}(x)=\lim_{\delta \rightarrow 0} \inf (F(v)-F(u)) /(v-u)$, where the infimum extends over $u$ and $v$ such that $u<x<v$ and $v-u<\delta$. Define $F^{*}(x)$ as this limit with the infimum replaced by a supremum. Show that in Theorem 31.4, $F^{\prime }$ can be replaced by $F^{*}$ in part (i) and by $F_{*}$ in part (ii). Show that in Theorem 31.6, ${ }_{F} D$ can be replaced by $F_{*}$ (note that $F_{*}(x) \leq { }_{F} D(x)$ ).

Lebesgue's density theorem. A point $x$ is a density point of a Borel set $A$ if $\lambda ((u, v] \cap A) /(v-u) \rightarrow 1$ as $u \uparrow x$ and $v \downarrow x$. From Theorems 31.2 and 31.4 deduce that almost all points of $A$ are density points. Similarly, $\lambda ((u, v] \cap A) /(v-u) \rightarrow 0$ almost everywhere on $A^{c}$.

Let $f:[a, b] \rightarrow R^{k}$ be an aic; $f(t)=\left(f_{1}(t), \ldots , f_{k}(t)\right)$. Show that the arc is rectifiable if and only if each $f_{i}$ is of bounded variation over $[a, b]$.

$\uparrow$ Suppose that $F$ is continuous and nondecreasing and that $F(0)=0$, $F(1)=1$. Then $f(x)=(x, F(x))$ defines an arc $f:[0,1] \rightarrow R^{2}$. It is easy to see by monotonicity that the arc is rectifiable and that, in fact, its length satisfies $L(f) \leq 2$. It is also easy, given $\epsilon$, to produce functions $F$ for which $L(f)>2-\epsilon$. Show by the arguments in the proof of Theorem 31.4 that $L(f)=2$ if $F$ is singular.

Suppose that the characteristic function of $F$ satisfies $\limsup_{t \rightarrow \infty }\left|\varphi (t)\right|=1$. Show that $F$ is singular. Compare the lattice case (Problem 26.1). Hint: Use the Lebesgue decomposition and the Riemann-Lebesgue theorem.

Suppose that $X_{1}, X_{2}, \ldots$ are independent and assume the values $\pm 1$ with probability $\frac{1}{2}$ each, and let $X=\sum_{n=1}^{\infty } X_{n} / 2^{n}$. Show that $X$ is uniformly distributed over $[-1,+1]$. Calculate the characteristic functions of $X$ and $X_{n}$ and deduce (1.40). Conversely, establish (1.40) by trigonometry and conclude that $X$ is uniformly distributed over $[-1,+1]$.

31.2 ↑ Show that the Cantor function is the distribution function of $\sum_{n=1}^{\infty } X_{n} / 3^{n}$, where the $X_{n}$ are independent and assume the values 0 and 2 with probability $\frac{1}{2}$ each. Express its characteristic function as an infinite product.

Show for the $F$ of Example 31.1 that ${ }_{F} D(1)=\infty$ and $D^{F}(0)=0$ if $p_{0}<\frac{1}{2}$. From (31.17) deduce that ${ }_{F} D(x)=\infty$ and $D^{F}(x)=0$ for all dyadic rationals $x$. Analyze the case $p_{0}>\frac{1}{2}$ and sketch the graph

$6.14 \uparrow$ Let $F$ be as in Example 31.1, and let $\mu$ be the corresponding probability measure on the unit interval. Let $d_{n}(x)$ be the $n$th digit in the nonterminating binary expansion of $x$, and let $s_{n}(x)=\sum_{k=1}^{n} d_{k}(x)$. If $I_{n}(x)$ is the dyadic interval of order $n$ containing $x$, then

van der Waerden's continuous, nowhere differentiable function is $f(x)= \sum_{k=0}^{\infty } a_{k}(x)$, where $a_{0}(x)$ is the distance from $x$ to the nearest integer and $a_{k}(x)=2^{-k} a_{0}\left(2^{k} x\right)$. Show by the Weierstrass $M$-test that $f$ is continuous. Use (31.8) and the ideas in Example 31.1 to show that $f$ is nowhere differentiable. ${ }^{\dagger }$ A Lipschitz condition of order $\alpha$ holds at $x$ if $F(x+h)-F(x)=O\left(|h|^{\alpha }\right)$ as $h \rightarrow 0$, for $\alpha >1$ this implies $F^{\prime }(x)=0$, and for $0<\alpha <1$ it is a smoothness condition stronger than continuity and weaker than differentiability.

Show (see (31.31)) that (apart from addition of constants) a function can have only one representation $F_{1}+F_{2}$ with $F_{1}$ absolutely continuous and $F_{2}$ singular.

Show that the $F_{\mathrm{s}}$ in the Lebesgue decomposition can be further split into $F_{\mathrm{d}}+F_{\mathrm{cs}}$, where $F_{\mathrm{cs}}$ is continuous and singular and $F_{\mathrm{d}}$ increases only in jumps in the sense that the corresponding measure is discrete. The complete decomposition is then $F=F_{\mathrm{ac}}+F_{\mathrm{cs}}+F_{\mathrm{d}}$.

$14.4 \uparrow$ If $f$ is nonnegative and Lebesgue integrable, then by Theorem 31.3 and (31.8), except for $x$ in a set of Lebesgue measure 0,

if $u \leq x \leq v, u<v$, and $u, v \rightarrow x$. There is an analogue in which Lebesgue measure is replaced by a general probability measure $\mu$ : If $f$ is nonnegative and integrable with respect to $\mu$, then as $h \downarrow 0$,

on a set of $\mu$-measure 1. Let $F$ be the distribution function corresponding to $\mu$, and put $\varphi (u)=\inf [x: u \leq F(x)]$ for $0<u<1$ (see (14.5)). Deduce (31.36) from (31.35) by change of variable and Problem 14.4.