Published January 1, 2024
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Almost everywhere continuity of conditional expectations
Description
A necessary and sufficient condition on a sequence ${\{{\mathcal{A}_{n}}\}_{n\in \mathbb{N}}}$ of σ-subalgebras which assures convergence almost everywhere of conditional expectations for functions in ${L^{\infty }}$ is given. It is proven that for $f\in {L^{\infty }}(\mathcal{A})$ \[ \mathsf{E}(f|{\mathcal{A}_{n}})\stackrel{a.e.}{\longrightarrow }\mathsf{E}(f|{\mathcal{A}_{\mu a.e.}}).\]
Translated Descriptions
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This is an automatic machine translation with an accuracy of 90-95%
Translated Description (Arabic)
يتم إعطاء شرط ضروري وكافٍ على تسلسل ${\{{\ mathcal{A }_{ n }}\}_{ n\ in \mathbb{N }}}$ من σ - subalgebras الذي يضمن التقارب في كل مكان تقريبًا للتوقعات الشرطية للوظائف في ${ L^{\infty }}$. ثبت أنه مقابل $f\in {L^{\infty }} (\mathcal{A}) $\[\ mathsf {E}(f |{\mathcal{A }_{ n}})\stackrel{a.e }{\longrightarrow}\ mathsf {E}(f |{\mathcal{A }_{\mu a.e}}).\]Translated Description (French)
Une condition nécessaire et suffisante sur une séquence ${\{{\mathcal{A}_{n}}\}_{n\in \mathbb{N}}}$ de σ-subalgebras qui assure la convergence presque partout des attentes conditionnelles pour les fonctions dans ${ L^{\infty }}$ est donnée. Il est prouvé que pour $f\in {L^{\infty }}(\mathcal{A})$ \[ \mathsf{E}(f|{\mathcal{A}_{n}})\stackrel{a.e.}{\longrightarrow }\mathsf{E}(f|{\mathcal{A}_{\mu a.e.}}).\]Translated Description (Spanish)
Se da una condición necesaria y suficiente en una secuencia ${\ {{\mathcal{A}_{n}}\}_{n\in \mathbb{N}}}$ de σ-subálgebras que asegura la convergencia en casi todas partes de las expectativas condicionales para las funciones en ${L^{\infty }}$. Está comprobado que para $f\in {L^{\infty }}(\mathcal{A})$ \[ \mathsf{E}(f|{\mathcal{A}_{n}})\stackrel{a.e.}{\longrightarrow }\mathsf{E}(f|{\mathcal{A}_{\mu a.e.}}).\]Files
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Additional details
Additional titles
- Translated title (Arabic)
- في كل مكان تقريبًا، استمرارية التوقعات المشروطة
- Translated title (French)
- Presque partout continuité des attentes conditionnelles
- Translated title (Spanish)
- En casi todas partes, continuidad de las expectativas condicionales
Identifiers
- Other
- https://openalex.org/W4390760918
- DOI
- 10.15559/23-vmsta240
References
- https://openalex.org/W2001137574
- https://openalex.org/W2016315504
- https://openalex.org/W2018120868
- https://openalex.org/W2040649804
- https://openalex.org/W2284153224
- https://openalex.org/W2962877653
- https://openalex.org/W3007301222