\documentclass[11pt]{article} \input{listki-mera.tex} \begin{document} \listok{2}{Lebesgue measure} \subsection{Boolean algebras} \defn {\bf Lattice} is a set $L$ endowed with algebraic binary operations $\wedge$ and $\vee:\; L\times L \arrow L$, called respectively {\bf meet} and {\bf join}, satisfying the following: \begin{enumerate} \item idempotent laws: $a\wedge a = a \vee a =a$. \item commutativity: $a\wedge b = b\wedge a, a\vee b = b\vee a$. \item associativity: $a\wedge (b\wedge c) = (a\wedge b)\wedge c$, $a\vee (b\vee c) = (a\vee b)\vee c$, \item absorbtion laws: $a \vee (a\wedge b) = a$, $a \wedge (a\vee b) = a$. \end{enumerate} \edefn \problem \label{_chasti_upo_reshe_Zadacha_} Let $(S, \preceq)$ be a poset such that for any $x, y\in S$ there exists {\bf supremum} $t\in S$, denoted $t:=x\vee y$ (i.e. such an element $t\succeq x, y$ that any $z$ satisfying $z\succeq x, y$ satisfies $z\succeq t$), and {\bf infimum} $u\in S$, denoted $u:=x\wedge y$, (i.e. such $t\preceq x, y$ that any $z$ satisfying $z\preceq x, y$ satisfies $z\preceq u$). Show that it is a lattice. \eproblem \problem[!] Let $L$ be a lattice. Introduce on $L$ a binary relation $x\preceq y$, that holds true whenever $x\wedge y = x$. \begin{enumerate} \item Show that $x\preceq y$ iff $x\vee y = y$. \item Show that $x\preceq y$ is a partial order relation. \item Consider $(L, \preceq)$ as a poset. Show that it has supremum and infimum. Show that they can be expressed as $(x\vee y)$, resp. $(x \wedge y)$. \item Show that any lattice can be obtained from a poset using the construction described in Problem~\ref{_chasti_upo_reshe_Zadacha_}. \end{enumerate} \eproblem \problem Let $R$ be a factorial ring. Construct a lattice from it, using operations of taking the least common multiple and of taking the greatest common divisor. \eproblem \problem Consider the following partial order relation on $2^S$: $ x\preceq y$, if $x\subset y$. Show that $(2^S, \preceq)$ have supremum and infimum. Show that they correspond to intersections and unions of sets. \eproblem \defn Boolean algebra is a construction to axiomatise the operations of intersection and union in the algebra of subsets. Boolean algebras are named after an English mathematician George Bool, 1815-1864. \hfill {\bf Boolean algebra} $(A,\vee, \wedge)$ is a lattice satisfying the following conditions: \begin{enumerate} \item Boundedness from below: $A$ contains an element $0$ such that $x\wedge 0=0$. \item Boundedness from above: $A$ contains an element $1$ such that $x\vee 1=1$. \item Distributivity: $(a\vee b)\wedge c = (a\wedge c) \vee (a\wedge c)$ \item Existence of complements: for any $x\in A$ there exists $\neg x$ such that $x\wedge \neg x=0$, $x\vee \neg x=1$. \end{enumerate} \edefn \problem Show that 0, 1, $\neg x$ are uniquely determined by the lattice structure on $A$. \eproblem \problem Show that $\neg 0=1$, $\neg 1=0$. \eproblem \problem Prove de Morgan laws: $\neg(a\vee b) = (\neg a) \wedge (\neg b)$, $\neg(a\wedge b) = (\neg a) \vee (\neg b)$. \eproblem \problem (Boolean algebra duality) Let $(A,\vee, \wedge)$ be a Boolean algebra. Consider operations $\vee_1:=\wedge$, $\wedge_1:=\vee$. Show that $(A,\wedge_1, \vee_1)$ is a Boolean algebra, too. \eproblem \problem Construct a two-element Boolean algebra. \eproblem \problem \label{_idempo_boolean_Zadacha_} Let $R$ be a commutative ring, and $V$ its set of idempotents (i.e. $a\in R$ satisfying $a^2=a$). Consider operations $e\vee f= e+f-ef$, $e\wedge f= ef$. Show that they give a Boolean algebra structure on $V$. \eproblem \defn {\bf Symmetric difference} in a Boolean algebra is defined by setting $a\triangle b:= (a\vee b) \wedge \neg (Á\wedge b)$ \edefn \problem[!] \begin{enumerate} \item Show that the symmetric difference is associative. \item Show that $\wedge$ is distributive w.r.t. $\triangle$. \item SHow that $(A, \wedge, \triangle)$ is a ring, where addition is given by $\triangle$ and multiplication by $\wedge$. \item Show that all the elements of this ring are idempotent. \end{enumerate} \eproblem \problem[!] Let $R$ be a commutative ring over $\Z/2\Z$, with all its elements idempotent. Consider the structure of Boolean algebra on $R$, defined in Problem \ref{_idempo_boolean_Zadacha_}. Show that $R$ can be obtained from this Boolean algebra by the construction described above. \eproblem \defn {\bf Ideal} in a Boolean algebra is a subset $I\subset A$, closed w.r.t. $\vee$, that satisfies $a\wedge i\in I$ for any $a\in A, i\in I$. \edefn \problem Show that a Boolean algebra with more than 2 elements contains a nontrivial ideal. \eproblem \problem[!] Let $(A,\wedge, \vee)$ be Boolean algebra with an ideal $I\subset A$. On $A$, define the relation $a\sim_I b := a\triangle b\in I$. Show that $\sim_I$ is an equivalence relation. Show that $\wedge$ and $\vee$ preserve the corresponding equivalence classes, and induce on the set $A'$ of the classes the structure of a Boolean algebra. \eproblem \defn Under these operations, $A'$ is called a {\bf quotient algebra (modulo $I$)}, and is denoted by $A/I$. The ideal $I$ is called {\bf maximal} when $A/I$ consists of 2 elements. \edefn \problem[*] Show that any nontrivial ideal of a Boolean algebra is contained in a maximal ideal. \eproblem \defn {\bf Representation}, or {\bf injective representation} of a Boolean algebra $A$ is an injective homomorphism $A\arrow 2^S$, defined for a set $S$. In orther words, a representation of $A$ is its realisation as a (sub)algebra of sets. \edefn \problem[*] \begin{enumerate} \item Show that each Boolean algebra admits an injective representation. \item Let $A$ be a finite Boolean algebra. Show that $A$ cosists of $2^n$ elements. Show that $A$ is isomorphic to the algebra of all subsets of $S=\{1,\dots,n\}$. \end{enumerate} \eproblem \subsection{External measure} From now on $S$ will denote a set, and ${\goth U}\subset 2^S$ will denote a ring of subsets that contains $S$. (Such a ring is called {\bf algebra of subsets}, or {\bf subalgebra of subsets in $2^S$}). Consider $2^S$ as a Boolean algebra, with operations $\vee=\cup$ and $\wedge=\cap$. Obviously ${\goth U}$ is a Boolean subalgebra of $2^S$. Consider a function $\mu:\; {\goth U}\arrow \R \cup \{\infty\}$. On the set $\R \cup \{\infty\}$ an addition operation is defined, so that $x+\infty = \infty$ and $\infty + \infty = \infty$. \defn Function $\mu:\; {\goth U}\arrow \R \cup \{\infty\}$ is called a {\bf finitely additive measure}, if for any nonintersecting $A, B\in {\goth U}$ one has $\mu(A\coprod B) = \mu(A)+\mu(B)$. A measure is called {\bf nonnegative}, if $\mu(A)\geq 0$ holds for any $A$. \edefn \defn Under these assumptions, let $X\subset S$. Define {\bf external measure} $\mu^*(X)$ as \[ \mu^*(X):= \inf_{\{A_i\}}\sum \mu(A_i), \] where $\inf$ is taken over all countable collections $\{A_i\}\subset {\goth U}$ covering $X$. We say that $X$ is of {\bf measure $0$} if $\mu^*(X)=0$. Wer say that $\mu$ is $\sigma$-additive if $\mu^*(A)=\mu(A)$ for any $A\in {\goth U}$. \edefn \problem Show that $\mu^*(A\cup B) \leq \mu^*(A)+\mu^*(B)$. \eproblem \problem[*] Give an example of a nonadditive external measure (i.e. of a measure $\mu$ for which $\mu^*(A\coprod B) = \mu^*(A)+\mu^*(B)$ fails). \eproblem \problem[!] Let $A$ have measure 0. Show that $\mu^*(A\cup B)=\mu^*(B\backslash A) = \mu^*(B)$. \eproblem \problem[!] Show that a countable union of measure 0 sets has measure 0. \eproblem \problem Show that the measure 0 sets make up a a Boolean ideal in the Boolean algebra $2^S$. \eproblem \problem[*] Let ${\goth U}\subset 2^S$ be the algebra generated by intervals on a closed segment or the line, with the usual measure. Give an example of uncountable measure 0 subset. \eproblem \problem Consider a smooth diffeomorphism $\phi$ of closed segments (smooth on the endpoints, too). Show that $\phi$ maps sets of measure 0 to sets of measure 0. \eproblem \problem Let $\phi$ be a diffeomorphism from an interval to the line. Show that $\phi$ maps sets of measure 0 to sets of measure 0. \eproblem \subsection{Measurable sets} \defn Consider the measure 0 subsets as a Boolean ideal in the Boolean algebra $2^S$. If $\mu^*(A\triangle B)=0$ for $A, B \subset S$ then we say that {\bf $A$ and $B$ coincide almost everywhere}. The quotient algebra modulo the ideal of measure 0 subsets is called {\bf the algebra of subsets of $S$ modulo the measure 0 subsets}. In the remainder of this set of problems we denote this algebra by $2^S/\sim$. \edefn \problem Fix $x\in S$. Assume that $\{ x\}\in {\goth U}$. Let the measure of a subset $X\subset S$ is given by the following rule: $\mu(X)=1$ if $x\in X$, and $\mu(X)=0$ otherwise. Find ${\goth U}/\sim$. \eproblem \problem[!] Define a function $d:\; 2^S \times 2^S\arrow \R$ by $d(A, B):= \mu^*(A\triangle B)$. Show that $d$ satisfies the triangle inequality: $d(A,B)\leq d(A, C) + d(B, C)$. \eproblem \problem [!] Let $\mu^*(A_1\triangle A_2)=0$. Show that $\mu^*(A_1\triangle B)=\mu^*(A_2\triangle B)$, for any $B\in 2^S$. \eproblem \remark The latter problem implies that $d(A, B)= \mu^*(A\triangle B)$ is well-defined on $2^S/\sim$. \eremark \problem Show that the function $d(A, B)= \mu^*(A\triangle B)$ defines a metric on $2^S/\sim$. \eproblem \problem[!] Consider the completion of $2^S/\sim$ with respect to this metric. Show that it is a Boolean algebra. \eproblem \defn Let $\{X_i\}$ be a sequence of subsets of $S$. {\bf The inverse limit} of $\{X_i\}$ is the set \[ \lim\limits_{\leftarrow}\{X_i\}:= \bigcup_i \left(\bigcap_{j>i} X_j\right). \] \edefn \problem Show that a reordering of $\{X_i\}$ does not change the inverse limit. \eproblem \problem Let $A\in 2^S$ and $\{X_i\}\subset 2^S$, whereas $d(A, X_i) = \lambda_i$. Show that \[ d(A, \lim\limits_{\leftarrow}\{X_i\}) \leq \sum \lambda_i.\] \eproblem \problem[!] Let $\{X_i\}$ be a Cauchy sequence in $2^S/\sim$. Show that it converges to $\lim\limits_{\leftarrow}\{X_i\}$. \eproblem \hint Replacing $\{X_i\}$ by a subsequence, ensure that \[ d(X_i, X_j)< 2^{-\min(i,j)}.\] Using the previous problem, check that \[ d(X_i,\lim\limits_{\leftarrow}\{X_i\}) \leq \frac 1 {2^{i-1}}. \] \eu \defn The set $X\subset S$ is called {\bf measurable} if it lies in the completion of ${\goth U}/\sim$ with respect to the metric defined above. \edefn \problem[!] Show that the measureable sets for a subalgebra in $2^S$. \eproblem \problem[**] Using the Axiom of Choice, give an example of unmeasurable subset of $[0,1]$ (with the standard measure). \eproblem \problem[!] (Lebesgue Theorem) Show the finite additivity of the function $\mu^*$ on the measurable sets (i.e. $\mu^*(A\coprod B) = \mu^*(A)+\mu^*(B)$ holds). \eproblem \hint Use the fact that the aglebra of measurable sets is the completion of ${\goth U}/(\sim\cap {\goth U})$, and there $\mu$ is additive. \eu \defn Let $\mu$ be $\sigma$-additive. In this case the function $\mu^*$ on the algebra of measurable sets is called {\bf continuation} of $\mu$. We denote it by $\mu$, as well. \edefn \problem[!] Let $\{A_i\}\subset {\goth U}$ be a countable sequence of nonintersecting sets, such that the series $\sum \mu(A_i)$ converges. Show that $\bigcup A_i$ is measurable. \eproblem \problem[!] Show that on the measurable sets the function $\mu^*$ is {\bf countably additive}, i.e. satisfies $\mu(\coprod X_i)=\sum \mu(X_i)$ \eproblem \subsection{Lebesgue measure} \defn Let ${\goth W}\subset 2^S$ be an algebra of subsets. ${\goth W}$ is called {\bf $\sigma$-algebra} if it is closed w.r.t. taking countable unions: $\bigcup X_i$ belongs to ${\goth W}$ for any countable collection of subsets $\{X_i\}\subset {\goth W}$. \edefn \problem[!] Let ${\goth U}\subset 2^S$ be an algebra of subsets equipped with a countably additive and nonnegative measure $\mu:\; {\goth U}\arrow \R \cup\{\infty\}$. Assume that $\mu(S)<\infty$. Show that its algebra of measurable subsets is a $\sigma$-algebra. \eproblem \defn {\bf Measure} on a $\sigma$-algebra ${\goth W}\subset S$ is a countably additive, nonnegative function ${\goth W}\arrow \R \cup\{\infty\}$. \edefn \problem[*] Give an example of a finitely additive, but not countably additive, measure. Show that a measure is countably additive iff $\mu^*(A)=\mu(A)$ for any $A\in {\goth W}$.\footnote{I.e. countable additivity and $\sigma$-additivity are in fact the same thing.} \eproblem \problem[!] Let $S_1$, $S_2$ be sets equipped with algebras \[ {\goth U_i}\subset 2^{S_i}, \ \ i = 1,2 \] and finitely additive nonnegative measures \[ \mu_i:\; {\goth U_i}\arrow \R \cup\{\infty\}.\] Consider the subalgebra ${\goth U_1}\times {\goth U_2}$ in $2^{S_1\times S_2}$, generated by the subsets of the form $A_1\times A_2$, where $A_i\in {\goth U_i}$. On each such subset define \[ \mu(A_1\times A_2) := \mu_1(A_1)\mu_2(A_2). \] \begin{enumerate} \item Show that $\mu$ can be extended to a finitely additive nonnegative measure on the ring ${\goth U_1}\times {\goth U_2}$. \item[**] Show that this extension is $\sigma$-addtitive whenever each $\mu_i$ is $\sigma$-additive. \end{enumerate} \eproblem \defn Consider a subalgebra ${\goth U} \subset 2^{\R^n}$ generated by open sets of the form $I_1\times I_2\times \dots I_n$, where $I_k$ - segments, intervals, or half-open intervals. We will call such an algebra of sets {\bf an algebra generated by parallelepipeds}. Extend the function \[ \mu(I_1\times I_2\times ... I_n) \arrow \prod |I_k| \] to a finitely additive measure $\mu$ on ${\goth U}$. Let ${\goth M}$ denote the completion of ${\goth U}$ w.r.t. $d(A, B):= \mu^*(A\triangle B)$, i.e. the set of the measurable subsets corresponding to ${\goth U}$ and $\mu$. The elements of ${\goth M}$ are called {\bf Lebesgue measurable} and the extension of $\mu^*$ to ${\goth M}$ is called the {\bf Lebesgue measure}. \edefn \problem[!] \begin{enumerate} \item Show that the Lebesgue measure is $\sigma$-additive on an algebra generated by parallelepipeds. \item Show that each open subset of $\R^n$ is measurable. \end{enumerate} \eproblem \defn Let $M$ be a topological space. The elements of the $\sigma$-algebra generated by its open sets are called {\bf Borel subsets} of $M$. \edefn \problem[!] Show that the Borel subsets of $\R^n$ are Lebesgue measurable. \eproblem \problem[!] Show that for each measurable $A\subset \R^n$ there exists a Borel subset $B\subset \R^n$ satisfying $\mu(B\triangle A)=0$. \eproblem \problem[*] Let $V\subset B\subset \R^n$ be a subset of an open ball $B$. Show that $V$ is measurable iff $\mu^*(V)+\mu^*(B\backslash V)= \mu(B)$ \eproblem \end{document}