By A. Pelczynski

This publication surveys effects touching on bases and diverse approximation houses within the classical areas of analytical capabilities. It includes huge bibliographical reviews.

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10. research in 2001. However, in that article I did not employ the Boolean-valued model approach, and hence the line of exposition was quite diﬀerent. Interested readers can easily ﬁnd the earlier version on the web. I thank Costas Drossos, Ali Enayat, Martin Goldstern, and Matt Yurkewych for pointing me to references [14], [5], [7], and [15] respectively. I am deeply indebted to Matthew Wiener and especially Andreas Blass for their many prompt and patient answers to my dumb questions about forcing.

Regazzini, De Finetti’s contribution to probability and statistics, Statistical Science 11 (1996), 253-282. [6] P. J. Daniell, A general form of integral, Annals of Mathematics 19 (1918), 279-294. [7] E. Diecidue, P. Wakker, and M. Zeelenberg, Eliciting decision weights by adapting de Finetti’s betting-odds method to prospect theory, Journal of Risk and Uncertainty 34 (2007), 179-199. [8] L. E. Dubins, On everywhere-deﬁned integrals, Transactions of the American Mathematical Society 232 (1977), 187-194.

We have realized the dream of adjoining a new subset of M to obtain a larger model (remember that U is a subset of B and we have required B to be in M ). It is, of course, not clear that M -generic ultraﬁlters exist in general. However, if M is countable, then it turns out to be easy to prove the existence of M -generic ultraﬁlters; essentially, one just lists the dense sets and hits them one by one. 10 Fortunately for us, the idea sketched at the beginning of Section 5 relies on M being countable anyway.