8.1.E: Problemas en las Funciones Medibles y Elementales\((S, \mathcal{M})\)
- Page ID
- 113783
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Rellene todos los datos de prueba en Corolarios 2 y 3 y Teoremas 1 y 2.
Demostrar que\(\mathcal{P}^{\prime} \cap P^{\prime \prime}\) es como se afirma al final de la Definición 2.
Dado\(A \subseteq S\) y\(f, f_{m}: S \rightarrow\left(T, \rho^{\prime}\right), m=1,2, \ldots,\) vamos
\ [
H=A\ izquierda (f_ {m}\ fila derecha f\ derecha)
\]
y
\ [
A_ {m n} =A\ izquierda (\ rho^ {\ prime}\ izquierda (f_ {m}, f\ derecha) <\ frac {1} {n}\ derecha).
\]
Demostrar que
\(H=\bigcap_{n=1}^{\infty} \bigcup_{k=1}^{\infty} \bigcap_{m=k}^{\infty} A_{m n} ;\)
(i) (ii)\(H \in \mathcal{M}\) si todos\(A_{m n}\) están\(\mathcal{M}\) dentro y\(\mathcal{M}\) es un\(\sigma\) anillo.
[Pista:\(x \in H\) iff
\ [
(\ forall n) (\ existe k) (\ forall m\ geq k)\ quad x\ en A_ {m n}.
\]
¿Por qué?]
Hacer el Problema 3 para\(T=E^{*}\) y\(f=\pm \infty\) seguir\(H\).
[Pista: Si\(\left.f=+\infty, A_{m n}=A\left(f_{m}>n\right) \cdot\right]\)
\(\Rightarrow 4\). Let\(f: S \rightarrow T\) be\(\mathcal{M}\) -elementary on\(A,\) con\(\mathcal{M}\) un\(\sigma\) -ring en\(S .\) Mostrar lo siguiente.
(i)\(A(f=a) \in \mathcal{M}, A(f \neq a) \in \mathcal{M}\).
(ii) Si\(T=E^{*},\) entonces
\(A(f<a), A(f \geq a), A(f>a),\) y\(A(f \geq a)\)
están en\(\mathcal{M},\) también.
iii)\((\forall B \subseteq T) A \cap f^{-1}[B] \in \mathcal{M}\).
[Pista: Si
\ [
A=\ bigcup_ {i-1} ^ {\ infty} A_ {i}
\]
y\(\left.f=a_{i} \text { on } A_{i}, \text { then } A(f=a) \text { is the countable union of those } A_{i} \text { for which } a_{i}=a .\right]\)
Hacer problema\(4(\mathrm{i})\) para medible\(f\).
[Pista: Si\(f=\lim f_{m}\) para mapas elementales\(f_{m},\) entonces
\ [
H=A (f=a) =A\ izquierda (f_ {m}\ fila derecha a\ derecha).
\]
Expresar\(H\) como en Problema\(3,\) con
\ [
A_ {m n} =A\ izquierda (h_ {m} <\ frac {1} {n}\ derecha),
\]
donde\(h_{m}=\rho^{\prime}\left(f_{m}, a\right)\) es elemental. (¿Por qué?) Luego usa Problemas\(4(\text { ii) and } 3(\text { ii }) .]\)
\(\Rightarrow 6\). Dado\(f, g: S \rightarrow\left(T, \rho^{\prime}\right),\) let\(h=\rho^{\prime}(f, g),\) es decir,
\ [
h (x) =\ rho^ {\ prime} (f (x), g (x)).
\]
Demostrar que si\(f\) y\(g\) son elementales, simples o medibles en\(A,\) lo que es\(h .\)
[Pista: Argumenta como en el Teorema 1. Teorema de uso\(4 \text { in Chapter } 3, §15 .]\)
\(\Rightarrow 7\). \(\left. \text { A set }\left.B \subseteq\left(T, \rho^{\prime}\right) \text { is called separable (in } T\right) \text { iff } B \subseteq \overline{D} \text { (closure of } D\right)\)para un conjunto contable\(D \subseteq T\).
Demostrar que si\(f: S \rightarrow T\) es\(\mathcal{M}\) -medible on\(A,\) entonces\(f[A]\) es separable en\(T .\)
[Pista:\(f=\lim f_{m}\) para mapas elementales\(f_{m} ;\) digamos,
\ [
f_ {m} =a_ {m i}\ text {on} A_ {m i}\ in\ mathcal {M},\ quad i=1,2,\ ldots
\]
Vamos a\(D\) constar de todo\(a_{m \mathrm{i}}(m, i=1,2, \ldots) ;\) así\(D\) es contable (¿por qué?) y\(D \subseteq T\).
Verifica que
\ [
(\ forall y\ en f [A]) (\ existe x\ en A)\ quad y=f (x) =\ lim f_ {m} (x),
\]
con\(f_{m}(x) \in D .\) Por lo tanto
\ [
(\ forall y\ en f [A])\ quad y\ in\ overline {D},
\]
por Teorema\(3 \text { of Chapter } 3, §16 .]\)
\(\Rightarrow 8\). Continuando Problema\(7,\) probar que si\(B \subseteq \overline{D}\) y\(D=\left\{q_{1}, q_{2}, \ldots\right\},\) entonces
\ [
(\ forall n)\ quad B\ subseteq\ bigcup_ {i=1} ^ {\ infty} G_ {q_ {i}}\ left (\ frac {1} {n}\ right),
\]
[Pista: Si\(p \in B \subseteq \overline{D},\) alguno\(G_{p}\left(\frac{1}{n}\right)\) contiene algunos\(q_{1} \in D ;\) así
\ [
\ rho^ {\ prime}\ izquierda (p, q_ {i}\ derecha) <\ frac {1} {n},\ texto {o} p\ en G_ {q_ {i}}\ izquierda (\ frac {1} {n}\ derecha).
\]
Así
\ [
\ izquierda. (\ forall p\ in B)\ quad p\ in\ bigcup_ {i-1} ^ {\ infty} G_ {q_ {i}}\ izquierda (\ frac {1} {n}\ derecha)\ cdot\ derecha]
\]
Demostrar Corolarios 2 y 3 y Teoremas 1 y\(2,\) asumiendo que\(\mathcal{M}\) es un semiring solamente.
Hacer Problema 4 para\(\mathcal{M}\) -mapas simples, asumiendo que\(\mathcal{M}\) es solo un anillo.