Math Problems Vault

Problem VIII.2.1

(a)(b)

Let $f$ be an arbitrary (not necessarily measurable) function, defined and almost everywhere finite on an arbitrary set $E \subset \mathbb {R}^{m}$ . Prove that for any $\varepsilon >0$

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(a)

the function $x \mapsto g(x)=\sup \left\{ f(y) \mid y \in E,\left\| x-y\right\| <\varepsilon \right\}$ is the restriction to $E$ of some function measurable on $\mathbb {R}^{m}$ (possibly taking infinite values as well);

(b)

for $m=1$ the function

x \mapsto h(x)=\sup \left\{ f(y) \mid y \in E \cap (x, x+\varepsilon )\right\}

is the restriction to $E$ of some function measurable on $\mathbb {R}$ .

Problem VIII.2.2

(a)(b)

Let $f$ be an arbitrary function defined on $\mathbb {R}$ with finite values. Prove that

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(a)

the function $\overline{f^{\prime }}$ , where

\overline{f^{\prime }}(x)=\varlimsup _{y \rightarrow x} \frac{f(y)-f(x)}{y-x},

is measurable;

(b)

the set of points at which the function $f$ has a derivative is measurable.

Problem VIII.2.3

(a)(b)

Let $f$ be a function defined on $\mathbb {R}$ with finite values. Prove that

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(a)

the function $\overline{f^{\prime }}{ }_{+}$ , where $\overline{f^{\prime }}{ }_{+}(x)=\varlimsup_{y \rightarrow x+0} \frac{f(y)-f(x)}{y-x}$ , is measurable if the function $f$ is measurable;

(b)

if $E \subset \mathbb {R}$ is a set having, as does its complement, no interior points, and $f=\chi_{E}$ , then

\bar{f}_{+}^{\prime }(x)=\begin{cases} 0, & \text{ for } x \in E ; \\ +\infty , & \text{ for } x \notin E . \end{cases}

Thus, the function $\overline{f^{\prime }}+$ may be non-measurable if the function $f$ is non-measurable.

Problem VIII.2.4

Let $f \in \mathscr {L}^{0}(E),\left|f\right| \leqslant 1$ almost everywhere on $E \subset \mathbb {R}^{m}$ . Prove that if $\left|f\right|<1$ on a set of positive measure, then there exist functions $f_{1}, f_{2} \in \mathscr {L}^{0}(E)$ , not equivalent to each other, satisfying the conditions $\left|f_{1}\right| \leqslant 1,\left|f_{2}\right| \leqslant 1$ almost everywhere on $E$ , such that $f=\frac{1}{2}\left(f_{1}+f_{2}\right)$ .

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Problem VIII.2.5

Give an example of a monotone function defined on $\mathbb {R}$ that is not equivalent to a continuous function on any (nonempty) interval.

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Problem VIII.2.6

The Dirichlet function (equal to one at rational points and zero at irrational points of the real line) is discontinuous at every point, yet is equivalent to a continuous function. Does there exist a function from $\mathscr {L}^{0}(\left[0,1\right])$ such that any function equivalent to it on $\left[0,1\right]$ is discontinuous at every point of this interval?

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Problem VIII.2.7

(a)(b)(c)

Let $C$ be the Cantor set, $f$ the Cantor function (see problem III.3.14), and let $g(x)=\frac{1}{2}(x+f(x))(x \in [0,1])$ . Prove that

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(a)

the function $g$ is strictly increasing;

(b)

$\lambda (g(C))>0$ ;

(c)

there exists a (measurable!) set $H \subset C$ such that the set $g(H)$ is non-measurable.

Remark.

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Let $E \subset \mathbb {R}^{m}, f \in \mathscr {L}^{0}(E)$ . With the function $f$ one can associate two monotone functions $F_{f}, F^{f}$ , defined by the equalities

\begin{array}{ll} F_{f}(t)=\lambda _{m}(\mathbb {E}\left[f<t\right]) & \text{ for } t \in \mathbb {R}, \\ F^{f}(t)=\lambda _{m}(\mathbb {E}\left[\left|f\right|>t\right]) & \text{ for } t>0 . \end{array}

Here in the first case we assume that for all $t$ $\lambda_{m}(\mathbb {E}\left[f<t\right])<\infty$ , and in the second -- that $\lambda_{m}(\mathbb {E}\left[\left|f\right|>t\right])<\infty$ for all $t>0$ . Obviously, sometimes one of the functions $F_{f}, F^{f}$ , or neither of them, cannot be defined. However, if $\lambda_{m}(E)<\infty$ , then both of them make sense. It is important to note that in the case of summability of the function $\left|f\right|$ , or at least summability of some positive power of it, the function $F^{f}$ is defined even if the measure of $E$ is infinite. It is clear that the function $F_{f}$ is increasing, while $F^{f}$ is decreasing. The functions $F_{f}$ and $F^{f}$ are called the (increasing and decreasing, respectively) distribution functions of the function $f$ . Functions $f$ and $g$ having the same increasing distribution function are called equimeasurable (it is not assumed here that $f$ and $g$ are defined on the same set).

Problem VIII.2.8

Prove that the increasing distribution function is left-continuous. When is it continuous at a point $t_{0} \in \mathbb {R}$ ?

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Problem VIII.2.9

(a)(b)(c)(d)

Find the increasing distribution functions on $\left[0,2 \pi \right]$ of the functions

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(a)

$\sin x$ ;

(b)

$\cos x$ ;

(c)

$\sin \frac{3 x}{2}$ ;

(d)

$\sin \left(2 x-\frac{\pi }{4}\right)$ .

Problem VIII.2.10

(a)(b)

Are the functions $\sin x$ and $\sin (n x+\alpha )(n \in \mathbb {N}, \alpha \in \mathbb {R})$ equimeasurable on the sets

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(a)

$\left[0,2 \pi \right]$ ;

(b)

$\left[0, \pi \right]$ ?

Problem VIII.2.11

(a)(b)

Let $F$ be the increasing distribution function of a function $f \in \mathscr {L}^{0}([0, T)), C, x_{0} \in \mathbb {R}$ .

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(a)

Find the increasing distribution functions of the functions $f+C$ , $C f(C>0), f^{3}$ .

(b)

Let $\widetilde{f}$ be the extension of the function $f$ to the set $\mathbb {R}$ with period $T$ , $g(x)=\widetilde{f}\left(x-x_{0}\right) \quad (x \in [0, T))$ . Find the increasing distribution function of the function $g$ .

Problem VIII.2.12

(a)(b)(c)

Let $E \subset \mathbb {R}^{m}, \lambda_{m}(E)=1, f \in \mathscr {L}^{0}(E)$ .

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(a)

Prove that there exists a unique number $t_{0}$ such that $\lambda_{m}\left(E\left(f \geqslant t_{0}\right)\right) \geqslant \frac{1}{2}$ and $\lambda_{m}\left(E\left(f \geqslant t_{0}+\varepsilon \right)\right)<\frac{1}{2}$ for any $\varepsilon >0$ .

(b)

Prove that $\lambda_{m}\left(E\left(f \leqslant t_{0}\right)\right) \geqslant \frac{1}{2}$ .

(c)

A number $M$ satisfying the inequalities

\lambda _{m}(E(f \geqslant M)) \geqslant \frac{1}{2}, \quad \lambda _{m}(E(f \leqslant M)) \geqslant \frac{1}{2},

is called a median of the function $f$ . Is the median unique?

Problem VIII.2.13

(a)(b)

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(a)

Prove that the functions $f$ and $f^{*}$ are equimeasurable. Assuming that the distribution function $F_{f}$ of the function $f$ is strictly increasing and continuous, find out how $F_{f}$ and $f^{*}$ are related.

(b)

Prove that if the function $f$ is concave (convex) on a finite interval, then so is the function $f^{*}$ .

Remark.

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Let $E \subset \mathbb {R}^{m}, f \in \mathscr {L}^{0}(E)$ . Set

f^{*}(\tau )=\inf \left\{ t \mid \lambda _{m}(E(f>t)) \leqslant \tau \right\} \quad \text{for } \tau \geqslant 0 .

The function $f^{*}$ is called the non-increasing rearrangement of the function $f$ (Fig. 12).

Fig. 12 Fig. 12

Problem VIII.2.14

(a)(b)(c)

Find the non-increasing rearrangements of the following functions:

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(a)

$\sin x$ on $\left[0,2 \pi \right]$ ;

(b)

$\sin \frac{x}{2}$ on $\left[0,2 \pi \right]$ ;

(c)

$\operatorname {tg} x$ on $(0, \pi )$ .

Remark.

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Let $E \subset \mathbb {R}^{m}, f \in \mathscr {L}^{0}(E)$ . The function $f^{*}(\tau )=\inf \left\{ t \mid \lambda_{m}(E(f>t)) \leqslant \tau \right\}$ , $\tau \geqslant 0$ , is called the non-increasing rearrangement of the function $f$ .

Problem VIII.2.15

Find the non-increasing rearrangement of the function

f(x, y)=x^{2}+y^{2} \quad \left(x^{2}+y^{2} \leqslant 1\right)

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Remark.

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Let $E \subset \mathbb {R}^{m}, f \in \mathscr {L}^{0}(E)$ . The function $f^{*}(\tau )=\inf \left\{ t \mid \lambda_{m}(E(f>t)) \leqslant \tau \right\}$ , $\tau \geqslant 0$ , is called the non-increasing rearrangement of the function $f$ .

Problem VIII.2.16

Let

f(x)=\sum _{1 \leqslant k \leqslant n} \frac{a_{k}}{x-c_{k}}, \quad \text{ where } \quad a_{1}>0, \ldots , a_{n}>0,

$c_{1}, \ldots , c_{n} \in \mathbb {R}$ . Prove that for any $t>0$

\lambda \left\{ x \in \mathbb {R} \mid f(x)>t\right\} =\frac{A}{t}, \quad \lambda \left\{ x \in \mathbb {R} \mid f(x)<-t\right\} =\frac{A}{t},

where $A=a_{1}+\ldots +a_{n}$ .

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Problem VIII.2.17

Let $E \subset \mathbb {R}^{m}, \lambda_{m}(E)=1, f \in \mathscr {L}^{0}(E)$ and let

0<A \leqslant \int _{E}\left|f(x)\right| d x, \quad \int _{E}\left|f(x)\right|^{2} d x \leqslant C^{2} A^{2}

for some $C \geqslant 1$ . Estimate from below the measure of the set

E_{A}=\left\{ x \in E \mid \left|f(x)\right|>\frac{A}{2}\right\} .

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Problem VIII.2.18

Let the function $f$ be $n$ times continuously differentiable on an interval $I$ and let $\left|f^{(n)}(x)\right| \geqslant \Delta >0$ for all $x \in I$ . Prove that

\lambda \left\{ x \in I \mid \left|f(x)\right| \leqslant t\right\} \leqslant 2 n^{2}\left(\frac{t}{\Delta }\right)^{1 / n} .

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