Based on the results of generalized additions, multiplications and differences proven in Part I and II of this paper a framework for a general integral is presented. Moreover it is shown that many results of the literature are contained as special cases in our results.
We establish a decomposition of non-negative Radon measures on which extends that obtained by Strichartz [6] in the setting of -dimensional measures. As consequences, we deduce some well-known properties concerning the density of non-negative Radon measures. Furthermore, some properties of non-negative Radon measures having their Riesz potential in a Lebesgue space are obtained.
In 1955, A. Revuz - Annales de l’Institut Fourier, vol. 6 (1955-56) - considered a type of Stieltjes measure defined on analogues of half-open, half-closed intervals in a partially ordered topological space. He states that these functions are finitely additive but his proof has an error. We shall furnish a new proof and extend some of this results to “measures” taking values in a topological abelian group.
An exponential inequality for Choquet expectation is discussed. We also obtain a strong law of large numbers based on Choquet expectation. The main results of this paper improve some previous results obtained by many researchers.
In this note we give a measure-theoretic criterion for the completeness of an inner product space. We show that an inner product space is complete if and only if there exists a -additive state on , the orthomodular poset of complete-cocomplete subspaces of . We then consider the problem of whether every state on , the class of splitting subspaces of , can be extended to a Hilbertian state on ; we show that for the dense hyperplane (of a separable Hilbert space) constructed by P. Pták and...