Math Problems Vault

Problem 105

Construct a subset of the line that is not Lebesgue measurable.

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Problem 106

Construct a Lebesgue measurable set in the plane whose projections on the coordinate axes are not measurable.

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Problem 107

Let $\mu$ be Lebesgue measure, and $X$ a measurable subset of $[0,1]$ . A point $x\in X$ is called a point of density of the set $X$ if

\lim _{\varepsilon \to 0}\frac{\mu \left(X\cap (x-\varepsilon ,x+\varepsilon )\right)}{2\varepsilon }=1.

Prove that almost all points of $X$ are points of density.

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Problem 108

Describe all the subsets $E$ of $[0,1]$ whose characteristic functions $\chi_E(x)$ are Riemann integrable.

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Problem 109

(a)(b)

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(a)

Let $X$ be a space with a $\sigma$ -additive measure. Prove that the subsets of zero measure in $X$ form a $\sigma$ -ring.

(b)

Prove that the countable sets on the line have Lebesgue measure zero. Give an example of an uncountable subset of the line having Lebesgue measure zero.

Problem 110

Prove that the set of all signed measures on a $\sigma$ -algebra $\mathfrak A$ is a linear space that is complete with respect to the distance

d(\nu _1,\nu _2)=\sup _{A\in \mathfrak A}\left|\nu _1\left(A\right)-\nu _2\left(A\right)\right|.

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Problem 111

For any subset $M$ of $\mathbb {R}^{n}$ let $M-M$ denote the set

M-M=\left\{ x-y:x\in M,y\in M\right\} .

Prove that if $M$ is measurable and has positive Lebesgue measure, then $M-M$ contains a neighborhood of zero in $\mathbb {R}^{n}$ .

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Problem 112

Let $X=\left\{ x_1,x_2,\ldots ,x_n,\ldots \right\}$ be a countable set, and assign to each element $x_i$ a number $p_i\geq 0$ in such a way that $\sum_{n=1}^{\infty }p_n=1$ . For any subset $A\subset X$ let $m(A)=\sum_{n\in N_A}p_n$ , where $N_A=\left\{ i:x_i\in A\right\}$ . Prove that $m$ is a $\sigma$ -additive measure on the algebra of all subsets of $X$ .

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Problem 113

Give an example of a finitely additive measure that is not $\sigma$ -additive.

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Problem 114

Compute the Wiener measure of the set of functions $f\in C[a,b]$ having the properties: $f(a)<0$ , $f(b)>0$ .

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Remark.

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The Wiener measure in §1.3 of Chapter II is the Lebesgue extension of the measure on the half-ring of cylinder sets

X(t_1,\ldots ,t_n;\Delta _1,\ldots ,\Delta _n)=\left\{ x\in C[a,b]:x(t_k)\in \Delta _k\text{ for }k=1,\ldots ,n\right\} ,

where $t_1\leq \cdots \leq t_n$ are points in $[a,b]$ and $\Delta_1,\ldots ,\Delta_n$ are intervals on the real line. Its values are

\begin{align} \mu \left(X(t_1,\ldots ,t_n;\Delta _1,\ldots ,\Delta _n)\right)={}& \pi ^{(1-n)/2}\prod _{k=1}^{n-1}(t_{k+1}-t_k)^{-1/2}\\ & \times \int _{\Delta _1}\cdots \int _{\Delta _n}\mathrm{exp}\left(-\sum _{k=1}^{n-1}\frac{(\tau _{k+1}-\tau _k)^2}{t_{k+1}-t_k}\right)\, d\tau _1\cdots d\tau _n. \end{align}

For $n=1$ , this means $\mu \left(X(t,\Delta )\right)=\int_\Delta d\tau =\left|\Delta \right|$ .

Problem 115

Define a measure $\mu$ on $[0,1]$ by the formula

\mu \left([\alpha ,\beta )\right)=\log _2\frac{1+\beta }{1+\alpha }.

Prove that this measure is preserved under the transformation $x\mapsto \left\{ 1/x\right\}$ , where $\left\{ \cdot \right\}$ denotes the fractional part of a number.

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Problem 116

(a)(b)

Each real number $x\in [0,1]$ can be expanded in a continued fraction

x=\cfrac 1{n_1+\cfrac 1{n_2+\cdots }}

(the finite fractions correspond to the rational numbers, and the infinite ones to the irrational numbers).

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(a)

Prove that in terms of the sequence $\left\{ n_k\right\}$ the transformation in Problem 115 has the form $\left\{ n_k\right\} \mapsto \left\{ n_{k+1}\right\}$ .

(b)

Problem 117

Prove that the condition of absolute convergence of the series $\sum \nu \left(A_n\right)$ in the definition of a signed measure can be replaced by the condition of simple convergence.

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Problem 118

(a)(b)

Compute the variation of the complex measure $\nu =\mu_1+i\mu_2$ on a set $A$ if:

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(a)

it is known that the measures $\mu_1$ and $\mu_2$ are disjoint on $A$ ;

(b)

$\mu_1=\mu_2$ on $A$ .

Problem 119

Let $\nu$ be a complex measure on a $\sigma$ -algebra $\mathfrak A\subset \mathcal P(X)$ . Prove that the real and imaginary parts of $\nu$ are signed measures on $\mathfrak A$ .

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Problem 120

Let $\nu$ be a signed measure on a $\sigma$ -algebra $\mathfrak A\subset \mathcal P(X)$ . Prove that $\sup_{A\in \mathfrak A}\nu \left(A\right)<+\infty$ , $\inf_{A\in \mathfrak A}\nu \left(A\right)>-\infty$ .

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Problem 121

Prove that supremum and infimum in Problem 120 are attained on some sets $A_+$ and $A_-$ in $\mathfrak A$ .

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Problem 122

In the notation of Problem 120 prove that the function $\nu$ (resp., $-\nu$ ) is a $\sigma$ -additive measure on $\mathfrak A\cap \mathcal P(A_+)$ (resp., on $\mathfrak A\cap \mathcal P(A_-)$ ).

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Problem 123

In the notation of Problem 120 prove that $\nu \left(A\right)=\nu \left(A\cap A_+\right)+\nu \left(A\cap A_-\right)$ for every $A\in \mathfrak A$ .

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Problem 124

Prove that the variation $\left|\nu \right|$ of a signed measure $\nu$ is finite and $\sigma$ -additive.

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