From a9912ce6c2879e0c1a02f1a06c5b8904b7dcb8e2 Mon Sep 17 00:00:00 2001 From: Robert Date: Tue, 30 Mar 2021 18:10:35 +0200 Subject: [PATCH] finished contents and measures --- chapters/sections/contents_measures.tex | 497 ++++++++++++++++++++++++ script.pdf | Bin 932121 -> 982677 bytes script.tex | 6 + 3 files changed, 503 insertions(+) diff --git a/chapters/sections/contents_measures.tex b/chapters/sections/contents_measures.tex index 61778cb..9240aca 100644 --- a/chapters/sections/contents_measures.tex +++ b/chapters/sections/contents_measures.tex @@ -11,4 +11,501 @@ \] A set $M$ is said to be countably infinite if it is countable and unbounded. \end{defi} + +\begin{rem} + \begin{enumerate}[(i)] + \item Countably infinite sets are the smallest kind of infinite sets. + \item Subsets of countable sets are countable. + \item The union of two countable sets is countable. + Let $\anyseqdef{M}, \anyseqdef[y]{K}$ by surjective sequences, then + \[ + (x_1, y_1, x_2, y_2, \cdots) + \] + is a surjective sequence for $M \subset K$. This argument can be used to prove $\intn$ is countable. + \item The union of countably many countable sets is countable. Let $M$ be a countable set of countable sets, + and $\anyseqdef[A]{M}$ a surjective sequence. Then $\forall n \in \natn$ exists a surjective mapping $(x_{n_k})_{k \in \natn} \subset A_n$ + + \begin{center} + \begin{tikzpicture} + \foreach \y in {1,...,3} + \foreach \x in {1,...,3} + \node at (\x, -\y) {$x_{{\y}_{\x}}$}; + + \foreach \y in {1,...,3} \node at (4, -\y) {$\cdots$}; + \foreach \x in {1,...,3} \node at (\x, -4) {$\vdots$}; + \node at (4, -4) {$\ddots$}; + + \node[above right, red] at (1, -1) {\tiny$1$}; + \node[above right, red] at (2, -1) {\tiny$2$}; + \node[above right, red] at (1, -2) {\tiny$3$}; + \node[above right, red] at (3, -1) {\tiny$4$}; + \node[above right, red] at (2, -2) {\tiny$5$}; + \node[above right, red] at (1, -3) {\tiny$6$}; + \end{tikzpicture} + \end{center} + This sequence is surjective on + \[ + \bigcup_{A \in M} A + \] + Especially, for countable $M, K$ we have + \[ + M \times K = \bigcup_{x \in M} \set[y \in K]{(x, y)} + \] + Thus $\natn \times \natn$, $\natn$, $\intn$ and $\ratn$ are countable. + + \item There exist uncountable sets, like $[0, 1]$, $\realn$ and $\powerset(\realn)$. + \end{enumerate} +\end{rem} + +\begin{defi} + Let $\Omega$ be a set. A family of subsets + \[ + (A_i)_{i \in I} \subset \powerset(\Omega) ~~(I \text{ denotes the index set}) + \] + is said to be pairwise disjoint is + \[ + A_i \cap A_j = \varnothing ~~\forall i, j \in I, ~i \ne j + \] +\end{defi} + +\begin{rem} + \begin{enumerate}[(i)] + \item Let $\setfam \subset \powerset(\realn^n)$ be a family of sets. A mapping + \[ + \mu: \setfam \rightarrow [0, \infty] + \] + is said to be the content of $\setfam$, if $\forall A_1, \cdots, A_k \in \setfam$ pairwise disjoint the following holds: + \[ + A_1 \cup \cdots \cup A_k \in \setfam \implies \mu(A_1 \cup \cdots \cup A_k) = \sum_{l=1}^k \mu(A_l) + \] + The content is a generalization of the concept of length ($\realn$), area ($\realn^2$), volume ($\realn^3$) etc. + + \item In the context of contents, measures and integrals we define + \begin{gather*} + c + \infty = \infty ~~\forall c \in \realn \cup \set{\infty} \\ + c \cdot \infty = \infty ~~\forall c \in (0, \infty] \\ + 0 \cdot \infty = 0 + \end{gather*} + + \item The goal is to choose the domain of the content as big as possible. Ideal would be $\setfam = \powerset{\realn^n}$. + This introduces the Banach-Tarski paradox: + \begin{itemize} + \item Let $\oball[1](0) \subset \realn^3$ be the unit sphere + \item One can show: There exists a disjoint decomposition + \[ + A_1 \cup \cdots \cup A_P \cup B_1 \cup \cdots \cup B_Q = \oball[1](0) + \] + and a set of translations and rotations + \[ + D_1, \cdots, D_P, \cdots T_1, \cdots, T_Q + \] + such that + \begin{align*} + D_1A_1 \cup D_2A_2 \cup \cdots \cup D_PA_P &= \oball[1](0) \\ + T_1B_1 \cup T_2B_2 \cup \cdots \cup T_QB_Q &= \oball[1](0) + \end{align*} + \end{itemize} + \end{enumerate} +\end{rem} + +\begin{defi} + Let $\Omega$ be a set, $\setfam$ a family of subsets of $\Omega$ (so $\setfam \subset \powerset(\Omega)$). $\setfam$ is sait to be a $\sigma$-algebra, if + \begin{enumerate}[(i)] + \item $\varnothing \in \setfam$ + \item $A \in \setfam \implies A^C = \Omega \setminus A \in \setfam$ + \item For a countable subset $\set{A_1, \cdots, A_n} \subset \setfam$ follows + \[ + \bigcup_{i \in \natn} A_i \subset \setfam + \] + \end{enumerate} + A mapping + \[ + \mu: \setfam \rightarrow [0, \infty] + \] + is said to be a measure, if + \[ + \mu\left(\bigcup_{i \in \natn} A_i \right) = \sum_{i \in \natn} \mu(A_i) ~~\text{ (} \sigma \text{-additivity)} + \] + for pairwise disjoint $(A_i)_{i \in \natn} \subset \setfam$ and $\mu(\varnothing) = 0$. + The pair $(\Omega, \setfam)$ is called a measureable space, and $(\Omega, \setfam, \mu)$ is called measure space. +\end{defi} + +\begin{eg} + \begin{enumerate}[(i)] + \item Let $\Omega$ be an arbitrary set, and let there be a disjoint decomposition + \[ + A_1 \cup \cdots \cup A_n = \Omega + \] + Then + \[ + \set[I \subset \set{1, \cdots, n}]{\bigcup_{i \in I} A_i} + \] + is a $\sigma$-algebra. + + \item Let $\Omega$ be arbitrary and $x \in \Omega$. Then + \begin{align*} + \delta_x: \powerset(\Omega) &\longrightarrow [0, \infty] \\ + A &\longmapsto \begin{cases} + 1, & x \in A \\ + 0, & x \notin A + \end{cases} + \end{align*} + is a measure. + + \item Let $\Omega$ be arbitrary, then + \begin{align*} + \#: \powerset(\Omega) &\longrightarrow [0, \infty] \\ + A &\longmapsto \begin{cases} + \text{Number of elements in } A, & A \text{ finite} \\ + \infty, & A \text{ infinite} + \end{cases} + \end{align*} + is the so called counting measure. It is useful for finite, countable sets. + + \item Let $\Omega$ be countable and $(a_w)_{w \in \realn} \subset [0, \infty]$. Then + \begin{align*} + \mu: \powerset(\Omega) &\longrightarrow [0, \infty] \\ + A &\longmapsto \sum_{w \in A} a_w + \end{align*} + a measure. + + \item Let $\measure$ be a measure space and $A \in \setfam$. Define the to $A$ confined $\sigma$-algebra + \[ + \setfam \vert_A := \set[B \in \setfam]{B \cap A} + \] + Then $(A, \setfam \vert_A, \mu)$ is a measure space. + \end{enumerate} +\end{eg} + +\begin{rem} + For countable subsets $\setfam = \set{A_1, \cdots, A_n, \cdots} \subset \sigma \text{-algebra}$ we have + \[ + \bigcap_{i \in \natn} A_i = \left(\bigcup_{i \in \natn} A_i^C\right)^C \subset \setfam + \] + If $A, B \in \setfam \implies A \setminus B \in \setfam$ then we can write + \[ + A \setminus B = A \cap B^C + \] + A measure $\mu$ is monotonic, which means if $A, B \in \setfam$ and $A \subset B$, then + \[ + \mu(B) = \mu(B \setminus A) + \mu(A) \ge \mu(A) + \] +\end{rem} + +\begin{defi} + A mapping $\mu: \powerset(\Omega) \rightarrow [0, \infty]$ is said to be an outer measure, if $\mu(\varnothing) = 0$ and + \[ + A \subset \bigcup_{i \in \natn} A_i \implies \mu(A) \le \sum_{i \in \natn} \mu(A_i) + \] + Just like measures, outer measures are monotonic. Let $\intervals$ be the family of bounded intervals, i.e. + \[ + \intervals = \bigcup_{\substack{x, y \in \realn \\x < y}} \set{[x, y], [x, y), (x, y], (x, y)} + \] + We define + \[ + l([x, y]) := l([x, y)) := l((x, y]) := l((x, y)) = y - x + \] +\end{defi} + +\begin{thm}\label{thm:outer} + The mapping + \begin{align*} + \lambda: \powerset(\realn) &\longrightarrow [0, \infty] \\ + A &\longmapsto \inf\set[A \subset \bigcup_{i \in \natn} I_i, ~I_i \in \intervals ~\forall i \in \natn]{\sum_{i=1}^{\infty} l(I_i)} + \end{align*} + defines an outer measure on the real numbers. Analogously one can create outer measures on $\realn^2, \realn^3$. +\end{thm} +\begin{proof} + We know + \begin{equation} + \lambda(\varnothing) \le l([0, \epsilon)) = \epsilon ~~\forall \epsilon > 0 + \end{equation} + which implies $\lambda(\varnothing) = 0$. We have to show that + \begin{equation} + A \subset \bigcup_{k \in \natn} A_k \implies \lambda(A) \le \sum_{k \in \natn} \lambda(A_k) + \end{equation} + If the right side is $\infty$ there is nothing to show. So let $\sum_{k \in \natn} \lambda(A_k) < \infty$. + Let $\epsilon > 0$, then $\forall k \in \natn ~~\exists (I_{k_i}) \subset \intervals$ such that + \begin{equation} + A_k \subset \bigcup_{i \in \natn} I_{k_i} \text{ and } \sum_{i \in \natn} l(I_{k_i}) \le \left(\lambda(A_k) + \frac{\epsilon}{2^k}\right) + \end{equation} + Then + \begin{equation} + A \subset \bigcup_{k = 1}^{\infty} A_k \subset \bigcup_{i, k \in \natn} I_{k_i} + \end{equation} + and + \begin{equation} + \lambda(A) \le \sum_{k, i \in \natn} l(I_{k_i}) \le \sum_{k \in \natn} \left(\lambda(A_k) + \frac{\epsilon}{2^k}\right) = \sum_{k \in \natn} \lambda(A_k) + \epsilon + \end{equation} + Since this inequality holds $\forall \epsilon > 0$ + \begin{equation} + \lambda(A) \le \sum_{k \in \natn} \lambda(A_k) + \end{equation} + must be true. The outer measure is not additive. +\end{proof} + +\begin{thm} + Let $\mu$ be an outer measure on $(\Omega, \powerset(\Omega))$. + Then the family of measureable sets + \[ + \setfam := \set[\mu(E) \ge \mu(E \cap A) + \mu(E \cap A^C) ~~\forall E \in \powerset(\Omega)]{A \subset \Omega} + \] + is a $\sigma$-algebra, and $\mu\vert_A$ a measure. +\end{thm} +\begin{thm} + Firstly, we always have + \begin{equation} + \mu(E) \le \mu(E \cap A) + \mu(E \cap A^C) + \end{equation} + which means $A$ is measureable if and only if + \begin{equation} + \mu(E) = \mu(E \cap A) + \mu(E \cap A^C) ~~\forall E \in \powerset(\Omega) + \end{equation} + It's easy to see that $\varnothing$ is measurable, and that + \begin{equation} + A \text{ measurable} \iff A^C \text{ measurable} + \end{equation} + We have + \begin{equation} + \begin{split} + E \cap (A \cup B) &= (E \cap A) \cup (E \cap B) \\ + &= (E \cap A) \cup (E \cap B \cap A^C) + \end{split} + \end{equation} + Which means that $\forall A, B$ measurable and $\forall E \in \powerset(\Omega)$: + \begin{equation} + \begin{split} + \mu(E) &= \mu(E \cap A) + \mu(E \cap A^C) \\ + &= \mu(E \cap A) + \mu(E \cap A^C \cap B) + \mu(E \cap A^C \cap B^C) \\ + &\ge \mu(E \cap (A \cup B)) + \mu(E \cap (A \cap B)^C) \ge \mu(E) + \end{split} + \end{equation} + So $A \cup B$ is measurable and it follows for disjoint $A, B$ + \begin{subequations} + \begin{align} + &\mu(E \cap A) + \mu(E \cap A^C \cap B) = \mu(E \cap (A \cup B)) \\ + \implies &\mu(E \cap A) + \mu(E \cap B) = \mu(E \cap (A \cup B)) \\ + \implies &\mu \text{ is additive for measurable sets} + \end{align} + \end{subequations} + Then by using induction we can see that finite unions of measurable sets are measurable and that for $A_1, \cdots, A_n$ measurable, pairwise disjoint sets + \begin{equation} + \mu\left(\bigcup_{i=1}^n A_i\right) = \sum_{i=1}^n \mu(A_i) + \end{equation} + holds. Now let $(A_i)_{i \in \natn}$ be pairwise disjoint measurable sets, and let + \begin{align} + S_n := \bigcup_{i=1}^n A_i && S := \bigcup_{i=1}^{\infty} A_i + \end{align} + Then $\forall E \in \powerset(\Omega)$ + \begin{equation} + \mu(E \cap S_n) = \sum_{i=1}^n \mu(E \cap A_i) + \end{equation} + To check measurability, consider + \begin{equation} + \begin{split} + \mu(E) &\ge \mu(E \cap S_n) + \mu(E \cap S_n^C) \\ + &\ge \sum_{i=1}^n \mu(E \cap A_i) + \mu(E \cap S^C) + \end{split} + \end{equation} + For $n \rightarrow \infty$: + \begin{equation} + \begin{split} + \mu(E) &\ge \sum_{i=1}^{\infty} \mu(E \cap A_i) + \mu(E \cap S^C) \\ + &\ge \mu\underbrace{(E \cap S)}_{\bigcup_{i=1}^{\infty} E \cap A_i} + \mu(E \cap S^C) \\ + &\ge \mu(E) + \end{split} + \end{equation} + Thus $S$ is measurable + \begin{equation} + \sum_{i=1}^{\infty} \mu(E \cap A_i) = \mu\left(E \cap \bigcup_{i=1}^{\infty} A_i\right) + \end{equation} + For $E = \Omega$ the $\sigma$-additivity follows. + It is left to show that for measurable (but not necessarily disjoint) $A_i$, that $\bigcup_{i=1}^{\infty} A_i$ is also measurable. + To do that define + \begin{equation} + B_i = A_i \setminus \left(\bigcup_{j=1}^{i-1} A_j \right) + \end{equation} + Then the $B_i$ are disjoint and measurable. Thus + \begin{equation} + \bigcup_{i=1}^{\infty} B_i = \bigcup_{i=1}^{\infty} A_i + \end{equation} + is measurable. +\end{thm} + +\begin{defi} + Application of the previous theorem on the outer measure from $\Cref{thm:outer}$ gives us the $\sigma$-algebra + of Lebesgue-measurable sets and the Lebesgue-measure $\lambda$. +\end{defi} + +\begin{rem} + $A \subset \realn$ is said to be a null set if its outer measure is $0$. Obviously + \[ + \lambda(\set{0}) = 0 + \] + For countable $A$ we have + \[ + \lambda(A) = \lambda\left(\cup_{x \in A} \set{x}\right) \le \sum_{x \in A} \lambda(\set{x}) = 0 + \] + So $\natn$, $\intn$ and $\ratn$ are null sets. Null sets are measurable, because + \[ + \forall E \in \powerset(\realn): ~~\underbrace{\lambda(E \cap A)}_0 + \lambda(E \cap A^C) = \lambda(E \cap A^C) \le \lambda(E) + \] +\end{rem} + +\begin{thm} + Intervals are Lebesgue measurable and + \[ + \lambda([a, b]) = b - a + \] +\end{thm} +\begin{proof} + Let $A$ be a bounded interval. Decompose $\realn$ into the intervals + \begin{equation} + \realn = I_L \cup A \cup I_R + \end{equation} + For $I \in \intervals$ we have $I \cap I_L, ~I \cap A, ~I \cap I_R$ bounded (or empty) intervals. + Now let $E \subset \powerset(\realn)$ and + \begin{equation} + E \subset \bigcup_{i \in \natn}I_i + \end{equation} + a covering. Then + \begin{align} + E \cap A \subset \bigcup_{i \in \natn} I_i \cap A && E \cap A^C \subset \bigcup_{i \in \natn} \left((I_i \cap I_L) \cup (I_i \cap I_R)\right) + \end{align} + are coverings of countably many intervals, and we have + \begin{equation} + \begin{split} + \sum_{i \in \natn} l(I_i) &= \sum_{i \in natn} l(I_i \cap A) + \sum_{i \in \natn} \left(l(I_i \cap I_L) + l(I_i \cap I_R)\right) \\ + &\ge \lambda(E \cap A) + \lambda(E \cap A^C) + \end{split} + \end{equation} + $\lambda$ is the infimum of all possible coverings + \begin{equation} + \lambda(E) \ge \lambda(E \cap A) + \lambda(E \cap A^C) + \end{equation} + And thus $A$ is measurable. + It is left to show that + \begin{equation} + A = [a, b] \implies \lambda(A) = b - a + \end{equation} + So let $\anyseqdef[I]{\intervals}$ such that + \begin{equation} + l = \sum_{n \in \natn} (I_n) < b - a + \end{equation} + First, let all $I_n$ be open. Choose + \begin{equation} + A_n = A \setminus \left(\bigcup_{i=1}^n I_i \right) + \end{equation} + Those $A_n$ are non-empty, since $A$ cannot be covered by finitely many intervals of length $< b - a$. + Choose a sequence $x_n \in A_n ~~\forall n \in \natn$. Since $A$ is a compact there exists a toward $x \in A$ convergent subsequence of $x_n$. + The point $x$ cannot be contained in any $I_n$, since because the $I_n$ are open, infinitely many $x_n$ would be contained in $I_n$, which would + contradict the construction of $A_n$. + \begin{equation} + \implies (I_n) \text{ do not cover } A + \end{equation} + For arbitrary $I_n$ (so not necessarily open), let $(x_k)$ be the sequence of the (countably many) boundary points of the intervals. + \begin{equation} + \epsilon = \frac{b - a - l}{4} > 0 + \end{equation} + And thus + \begin{equation} + \set[i \in \natn]{\interior{I}_i} \cup \set[\forall k \in \natn]{\left(x_k - \frac{\epsilon}{2^k}, x_k + \frac{\epsilon}{2^k}\right)} + \end{equation} + is a covering of $A$ by countably many open intervals of total length + \begin{equation} + \le l + \sum_{k = 1}^{\infty} \frac{2\epsilon}{2^k} = l + \frac{b - a - l}{2} = \frac{b - a + l}{2} < b - a + \end{equation} + which is impossible due to our construction above. +\end{proof} + +\begin{thm} + Open and closed sets are Lebesgue measurable. +\end{thm} +\begin{proof} + Let $O \subset \realn$ be open. It is to show that + \begin{equation} + O = \bigcup_{\substack{l, r \in \ratn \\ (l, r) \subset O}} (l, r) \implies O \text{ Lebesgue measurable} + \end{equation} + Let $x \in O$, since $O$ is open + \begin{equation} + \exists \epsilon > 0: ~~(x - \epsilon, x + \epsilon) \subset O + \end{equation} + Since $\ratn$ is dense in $\realn$ + \begin{equation} + \exists l, r \in \ratn: ~~x - \epsilon < l < x \text{ and } x < r < \epsilon + x + \end{equation} + So $x \in (l, r) \subset O$. + If $C$ is a closed set, then $\realn \setminus C$ is open and thus Lebesgue measurable. + \begin{equation} + \implies C = \realn \setminus(\realn \setminus C) \text{ Lebesgue measurable} + \end{equation} +\end{proof} + +\begin{rem} + The Lebesgue-$\sigma$-algebra contains many more sets. All sets that are "created by normal means" are Lebesgue measurable. +\end{rem} + +\begin{rem} + For $A \subset \realn$ and $x \in \realn$ we define + \[ + A + x := \set[y \in A]{y + x} + \] + A measure on $\realn$ is said to be invariant under translation, if + \[ + \mu(A) = \mu(A + x) ~~\forall A \in \setfam, ~x \in \realn + \] + Since translations of intervals result in intervals, the (outer) Lebesgue measure is invariant under translation. + One can show that the Lebesgue measure is the only translational symmetric measure on $\realn$, with + \[ + \lambda([0, 1]) = 1 + \] +\end{rem} + +\begin{thm} + Let $\measure$ be a measure space. For a monotonically increasing sequence $\anyseqdef[A]{\setfam}$ (this means $A_n \subset A_{n+1} ~~\forall n \in \natn$), we have + \[ + \mu\left(\bigcup_{n \in \natn} A_n\right) = \limn \mu(A_n) = \sup_{n \in \natn} \mu(A_n) + \] + For a monotonically decreasing sequence $\anyseqdef[B]{\setfam}$ we have + \[ + \mu\left(\bigcap_{n \in \natn} B_n\right) = \limn \mu(B_n) = \inf_{n \in \natn} \mu(B_n) + \] + if $\mu(B_N) < \infty$ for $N \in \natn$ +\end{thm} +\begin{proof} + If $\mu(A_n) = \infty$ for some $n \in \natn$ there is nothing to show. So let + \begin{equation} + \mu(A_n) < \infty ~~\forall n \in \natn + \end{equation} + Set $A_0 = \varnothing$ and define + \begin{equation} + C_n := A_n \setminus A_{n-1} + \end{equation} + These $C_n$ are pairwise disjoint, and thus + \begin{equation} + \begin{split} + \mu\left( \bigcup_{n \in \natn} A_n \right) &= \mu\left( \bigcup_{n \in \natn} C_n \right) = \sum_{n=1}^{\infty} \mu(C_n) = \underbrace{\sum_{n=1}^{\infty}\left( \mu(A_n) - \mu(A_{n-1}) \right)}_{\text{Telescoping series}} \\ + &= \limn \mu(A_n) - \underbrace{\mu(A_0)}_{= 0} + \end{split} + \end{equation} + Now let $\mu(B_N) < \infty \rightarrow \mu(B_n) < \infty ~~\forall n \ge N$. Set + \begin{equation} + D_n = B_N \setminus B_n ~~\forall n \ge N + \end{equation} + $\seq{D}$ is monotonically increasing and thus + \begin{equation} + \bigcup_{n = N}^{\infty} D_n = \bigcup_{n = N}^{\infty} B_N \cap B_n^C = B_N \cap {\underbrace{\left( \bigcap_{n = N}^{\infty} B_n \right)}_B}^C = B_N \cap B^C = B_N \setminus B + \end{equation} + Which in turn implies + \begin{equation} + \begin{split} + \mu(B_N) - \mu(B) &= \mu(B_N \setminus B) = \limn \underbrace{\mu(B_N \setminus B_n)}_{\mu(B_N) - \mu(B_n)} \\ + &= \mu(B_N) - \limn \mu(B_n) + \end{split} + \end{equation} +\end{proof} + +\begin{rem} + $\mu(B_N) < \infty$ for some $N \in \natn$ is a necessarily requirement.6 +\end{rem} \end{document} \ No newline at end of file diff --git a/script.pdf b/script.pdf index cd762b47c17c17787a3694aa4d98730c155b9c43..327c126def18741380d3c637bc3b8e78c022225f 100644 GIT binary patch delta 201106 zcmZsiWmH_t)~<1PcX#c^Ap{8;+}(pa1P{;<+?~dQ1t&lV?i$?P3Blbx+|J(TyGOEr 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