# Literature request: Functional capacities

Are the results of the following book (in French) covered in English in a book or in an article, and if so, could you please provide a reference?

C. Dellacherie, Ensembles analytiques, Capacités, Mesures de Hausdorff. Springer-Verlag, Berlin, 1972.

I'm interested in particular in the results on functional capacities on an abstract (paved) space. It appears that the results on capacities acting on sets are better documented in English.

Since capacities are such an interdisciplinary tool in Mathematics I thought this would be the right place to ask this question. Thank you very much in advance for your help!

Edit:

To be specific, I am interested in the analogue of Choquet's theorem for functional capacities. It appears to be impossible to find an English text that presents the theorem and its proof. In addition to Dellacherie's paper mentioned in my original message above I shall add the following paper by Gustave Choquet that supposedly touches the functional version of the capacitability theorem:

G. Choquet: Forme abstraite du théorème de capacitabilité. Ann. Inst. Fourier 9, 83–89 (1959)

• Update, just in case someone else stumbles upon to this same question: I personally found an appendix in the part C of the voluminous book of Dellacherie and Meyer the most helpful one. It is little confusing since the classical results on the descriptive set theory are covered already in the part A of the book! – Matti Kiiski May 24 '18 at 8:35

The paper, by C. Dellacherie himself,

C. Dellacherie, Capacities and analytic sets, Cabal seminar 77-79, Proc., Caltech-UCLA logic Semin. 1977-79, Lect. Notes Math. 839, 1-31 (1981).

covers, in part, the content from his french book.

Quoting Math Reviews 0611166 : These notes present analytic sets as definable through the notion of capacity. The route is dependent on the approximable character of analytic operations, a path explored by the author about ten years ago in his book [Ensembles analytiques, capacités, mesures de Hausdorff, Lecture Notes in Math., 295, Springer, Berlin, 1972]''

Concerning the applications to Hausdorff measures, it is clear from the comment section at the end of the book that the following three references should also be useful:

L. Carleson, Selected problems on exceptional sets. Van Nostrand Mathematical Studies, Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London 1967

C.A. Rogers, Hausdorff measures. Cambridge University Press, London, New York 1970

M. Sion, D. Sjerve, Approximation properties of measures generated by continuous set functions. Mathematika 9, 1962, 145-156.

• Thank you very much for these! It appears that these papers concentrate on capacities on sets. – Matti Kiiski Sep 21 '17 at 10:27