Looking for a proof of the (general) Salem-Zygmund theorem

Question

I'm looking for a proof of the general Salem-Zygmund theorem, as stated in page 38 of this paper.

I've been searching everywhere for a complete proof, but the only reference I've been able to find is the one given in that same paper, which is this, but sadly I don't read french.

Does anyone know where can I find a proof in english (or spanish)? Something reasonably self-contained would be ideal, or with a more abundant bibliography.

Answer 1

The result is proved starting on page 90 ($\S4$ of Chapter III) of

MR0953784 (90a:42008). Alexander S. Kechris and Alain Louveau. Descriptive set theory and the structure of sets of uniqueness. London Mathematical Society Lecture Note Series, 128. Cambridge University Press, Cambridge, 1987. viii+367 pp. ISBN: 0-521-35811-6.

(The link is to the review at MathSciNet by Howard Becker, it requires a subscription to MathReviews.)