Measure Theory

Prove that the symmetric difference operation satisfies the condition

(the analog of the triangle inequality for the set-valued ``distance'' $d(A,B)=A\triangle B$).

Prove that:

Show that a system of sets that is closed under the union and intersection operations is not necessarily a ring.

Prove that a system of sets that is closed under the union and difference operations is a ring.

Prove that the set of all intervals (open, closed, and half-open) on the line is a half-ring, but not a ring.

Show that for any nonempty system $S$ of sets there is exactly one minimal ring $R(S)$, i.e., a ring $R(S)$ of sets such that $S\subset R(S)$, and $R(S)\subset R$ for any ring $R$ containing $S$.

Prove that the minimal ring of a half-ring $S$ is the system of sets of the form $A=\coprod_{k=1}^{n}A_k$, $A_k\in S$.

Prove that any $\sigma$-algebra is a $\delta$-algebra and, conversely, any $\delta$-algebra is a $\sigma$-algebra.

The limit supremum of a sequence of sets $E_n$ is defined to be the set $\limsup_n E_n=\bigcap_n\left(\bigcup_{k\geq n}E_k\right)$, i.e., the collection of points belonging to infinitely many sets $E_n$. The limit infimum of a sequence of sets $E_n$ is defined to be the set $\liminf_n E_n=\bigcup_n\left(\bigcap_{k\geq n}E_k\right)$. Prove that $\liminf_n E_n\subset \limsup_n E_n$ for any sequence of sets $E_n$. If the limits supremum and infimum are equal, then their common value is called the limit of the sequence of sets $E_n$.

Give an example of a sequence of sets $E_n$ for which $\liminf_n E_n\neq \limsup_n E_n$.

Let $X$ be a set and $\left\{ E_n\right\}$ a sequence of sets such that $E_n\subset X$ for any $n$. Prove the formula

Let $\left\{ E_n\right\}$ be a sequence of sets, and $\left\{ \chi_n\right\}$ the sequence of their characteristic functions. Prove that the function $\limsup_n\chi_n$ is the characteristic function of the set $\limsup_n E_n$, and the function $\liminf_n\chi_n$ is the characteristic function of the set $\liminf_n E_n$.

Prove that the limit of a sequence of sets $E_n$ exists if and only if the sequence of characteristic functions of the $E_n$ has a limit.

Let $A$ be a system of sets and $\widetilde A$ the collection of characteristic functions of the sets in $A$. Prove that $A$ is a ring of sets if and only if $\widetilde A$ is an algebraic ring (with respect to addition and multiplication modulo 2).

The Borel sets on the line are defined to be the sets obtained from intervals by countably many operations of union, intersection, and difference. Prove that the collection of Borel sets has the cardinality of the continuum.

Prove that the cardinality of the set of Lebesgue measurable subsets of $[0,1]$ is greater than that of the continuum.

Let $\mu$ be Lebesgue measure on $[0,1]$, and introduce an equivalence relation on the space of Lebesgue measurable subsets of $[0,1]$ by setting $A\sim B$ if $\mu \left(A\triangle B\right)=0$. Prove that the collection of equivalence classes has the cardinality of the continuum.

Let $\mu$ be a measure on $S$. Prove that the following conditions are equivalent if $S$ is a ring, and may not be equivalent if $S$ is a half-ring.

1. Countable additivity: $\mu \left(\coprod_{k=1}^{\infty }A_k\right)=\sum_{k=1}^{\infty }\mu \left(A_k\right)$.

2. Upper semicontinuity: if $A_1\supset A_2\supset A_3\supset \cdots$ and $A=\bigcap_{k=1}^{\infty }A_k$, then $\mu \left(A\right)=\lim \mu \left(A_k\right)$.

3. Lower semicontinuity: if $A_1\subset A_2\subset A_3\subset \cdots$ and $A=\bigcup_{k=1}^{\infty }A_k$, then $\mu \left(A\right)=\lim \mu \left(A_k\right)$.

4. Continuity: $\mu \left(\lim_{n\to \infty }A_k\right)=\lim_{n\to \infty }\mu \left(A_k\right)$.

Let $\mu$ be a countably additive measure on a half-ring $S\subset \mathcal P(X)$, and $\mu^*$ the corresponding outer measure on $\mathcal P(X)$.

Let $S$ be the half-ring of intervals of the form $[a,b)$ on $[0,1]$, and $\mathcal M$ the space constructed in Problem 95(a). Prove that $\mathcal M$ is connected and not compact.

Let $\left\{ E_n\right\}$ be a sequence of Lebesgue measurable subsets of the line. Are the limits supremum and infimum of the sequence $\left\{ E_n\right\}$ measurable sets (see Problem 82)?

Let $A_n$ be a sequence of measurable sets with $\sum \mu \left(A_n\right)<\infty$. Prove that $\mu \left(\limsup A_n\right)=0$.

Suppose that $X$ is the unit square in the plane and $S$ is the half-ring of rectangles in $X$ of the form

Define $m(T_{ab})=b-a$. Describe the explicit form of the Lebesgue extension of the measure $m$.

Under the conditions and with the notation of Problem 100 prove that the set $\widetilde T=\left\{ 0\leq x\leq 1,\ y=1/2\right\}$ is not measurable, and find its outer measure.

Suppose that a measure $\mu$ is defined on a half-ring $X$ with unit, and let $\mu^*$ be its outer measure. A set $A\subset X$ is said to be measurable in the Carathéodory sense if for any subset $Z\subset X$

Prove that $A$ is Lebesgue measurable if and only if it is measurable in the Carathéodory sense.

Let $\mu$ be a $\sigma$-additive measure defined on a half-ring. A set $A$ is called a set of $\sigma$-uniqueness for $\mu$ if:

1. there exists a $\sigma$-additive extension $\lambda$ of $\mu$ that is defined on $A$;

2. for every two such $\sigma$-additive extensions $\lambda_1$ and $\lambda_2$, $\lambda_1\left(A\right)=\lambda_2\left(A\right)$.

Suppose that each of the sets $X_n$, $n=1,2,3,\ldots$, consists of the digits $0,1,2,\ldots ,9$. Define a measure $\mu_n$ on $X_n$ by setting $\mu_n\left(Y\right)=(1/10)\operatorname {card}Y$. Let the measure $\mu$ on $X=\prod X_n$ be the product of the measures $\mu_n$. Prove that $\mu$ passes into the usual Lebesgue measure on $[0,1]$ under the mapping $\left\{ x_n\right\} \mapsto 0.x_1x_2x_3\ldots$ (infinite decimal fraction) of $X$ into $[0,1]$.

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

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

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

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

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

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

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

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

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 $X$.

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

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

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

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.

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

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

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.

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

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 $\mathfrak A$.

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

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

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 $\mathfrak A\cap \mathcal P(A_-)$).

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 $A\in \mathfrak A$.

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