AbstractThis paper addresses the issue of finite versus countable additivity in Bayesian probability and decision theory – in particular, Savage’s theory of subjective expected utility and personal probability. I show that Savage’s reason for not requiring countable additivity in his theory is inconclusive. The assessment leads to an analysis of various highly idealized assumptions commonly adopted in Bayesian theory, where I argue that a healthy dose of, what I call, conceptual realism is often helpful in understanding the interpretational value of sophisticated mathematical structures employed in applied sciences like decision theory. In the last part, I introduce countable additivity into Savage’s theory and explore some technical properties in relation to other axioms of the system.
In the theory of spectral (and prespectral) operators in a Banach space or in a locally convex topological vector space the countable additivity (in some topology) of a resolution of the identity of the operator is a standing assumption. One might wonder why. Even if one cannot completely agree with the opinion of Diestel and Uhl ([6, p. 32]) stating that “countable additivity [of a set function] is often more of a hindrance than a help”, it might be interesting to study which portions of the theory of (pre)spectral operators and in which form extend to the more general situation described below.
AbstractX1 and X 2 are completely regular Hausdorff spaces, E 1, E 2 and F are Dedekind complete Banach lattices, 〈·,·〉: E 1 × E 2 → F is a bilinear mapping, and μ 1 and μ 2 are, respectively, E 1 and E 2 valued positive, countably additive Baire or Borel measures (countable additivity relative to order convergence) on X 1 and X 2. Under certain conditions the existence and uniqueness of the F-valued, positive, product measure is proved.
Countable additivity, idealization, and conceptual realism