8.11.E: Problemas en Derivados de Radón-Nikodym y Descomposición de Lebesgue
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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 Lema 2 y Teorema 1.
Verificar la declaración siguiendo la fórmula (3). Demostrar también lo siguiente:
(i) Si\(P \in \mathcal{M}\) junto con\(-P \in \mathcal{M},\) entonces\(s \perp t\) implica\(t \perp s\);
(ii)\(s \perp t\) iff\(v_{s} \perp t\).
Demostrar Corolario 1.
[Consejos: Aquí\(\mathcal{M}\) hay un\(\sigma\) anillo. Supongamos\(s\) y\(u\) residen en\(P^{\prime}\) y\(P^{\prime \prime},\) respectivamente, y\(v_{t} P^{\prime}=0=v_{t} P^{\prime \prime} .\) Let\(P=P^{\prime} \cup P^{\prime \prime} \in \mathcal{M} .\) Verify that\(v_{t} P=0\) (use Problema 8 en el Capítulo\(7, §11 \text { ). Then show that both } s \text { and } u \text { reside in } P .]\)
Demostrar que si\(s: \mathcal{M} \rightarrow E^{*}\) es una medida firmada en\(S \in \mathcal{M},\) entonces\(s^{+} \perp s^{-}\) y\(s^{-} \perp s^{+}\).
Rellenar todos los datos en el comprobante de Teorema\(2 .\) También acreditar lo siguiente:
(i)\(s^{\prime}\) y\(v_{s^{\prime}}\) son absolutamente\(t\) -continuos.
[Pista: Usar el Teorema 2 en el Capítulo 7, §11.]
ii)\(v_{s}=v_{s^{\prime}}+v_{s^{\prime \prime}}, v_{s^{\prime \prime}} \perp t\).
(iii) Si\(s\) es una medida\((\geq 0),\) así lo son\(s^{\prime}\) y\(s^{\prime \prime}\).
Verificar la Nota 3 para el Teorema 1 y el\(3 .\) Estado Corolario y probar ambas proposiciones generalizadas con precisión.