REPRESENTATIONS OF WEAK AND STRONG INTEGRALS IN BANACH SPACES | Zendy

James K. Brooks | Zendy

AI Assistant Blog Pricing

Home ZAIA Blog

Open Access

REPRESENTATIONS OF WEAK AND STRONG INTEGRALS IN BANACH SPACES

Author(s) -

James K. Brooks

Publication year - 1969

Publication title -

proceedings of the national academy of sciences

Language(s) - English

Resource type - Journals

eISSN - 1091-6490

pISSN - 0027-8424

DOI - 10.1073/pnas.63.2.266

Subject(s) - mathematics , absolute convergence , additive function , series (stratigraphy) , banach space , conditional convergence , representation (politics) , integrable system , pure mathematics , daniell integral , conditional expectation , convergence (economics) , order (exchange) , countable set , weak convergence , mathematical analysis , fourier integral operator , operator theory , fourier series , computer security , law , asset (computer security) , economic growth , computer science , biology , paleontology , political science , econometrics , finance , politics , economics

We establish a representation of the Gelfand-Pettis (weak) integral in terms of unconditionally convergent series. Moreover, absolute convergence of the series is a necessary and sufficient condition in order that the weak integral coincide with the Bochner integral. Two applications of the representation are given. The first is a simplified proof of the countable additivity and absolute continuity of the indefinite weak integral. The second application is to probability theory; we characterize the conditional expectation of a weakly integrable function.

The content you want is available to Zendy users.

Already have an account? Click here to sign in.

Having issues? You can contact us here

Accelerating Research