Almost everywhere continuity of conditional expectations
- Resource Type
- article
- Authors
- Alberto Alonso; Fernando Brambila-Paz
- Source
- Modern Stochastics: Theory and Applications, Vol 11, Iss 3, Pp 247-263 (2024)
- Subject
- Conditional expectations
probability
Applied mathematics. Quantitative methods
T57-57.97
Mathematics
QA1-939
- Language
- English
- ISSN
- 2351-6046
2351-6054
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 $f\in {L^{\infty }}(\mathcal{A})$ \[ \mathsf{E}(f|{\mathcal{A}_{n}})\stackrel{a.e.}{\longrightarrow }\mathsf{E}(f|{\mathcal{A}_{\mu a.e.}}).\]