Theorem 12 in Appendix A of Pollard's A User's Guide to Measure Theoretic Probability gives conditions under which a set function defined on a family of sets $\mathscr{K}$ which is closed under finite unions and intersections and contains the null set can be extended to a countably-additive measure on $\sigma(\mathscr{K})$ (note no conditions are imposed on the complements of sets in $\mathscr{K})$.

The usual textbook presentations of Carathéodory's Theorem assume that the initial collection over which the set function is defined is a field or a semiring. Also Pollard proceeds by approximating sets from the inside by elements of $\mathscr{K}_0$ while the usual presentations of Carathéodory's theorem approximate sets from the outside.

I would like to know whether there are any known concrete situations where the higher generality of the result given in Pollard and the method of building measures from the inside (an approach that Pollard attributes to Topsøe and seems to have been followed later by Konig) would be more useful than the standard approach.



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