I'm looking for a well-written, rigorous and self-contained treatment of multidimensional characteristic functions, specifically Lévy's continuity theorem and the uniqueness theorem (which states that distinct multi-dimensional distributions engender distinct characteristic functions).
In particular, all necessary results from real and complex analysis should be proved, or else specific references should be given to where proofs can be found. Preferably, it should not be assumed that the theory of one-dimensional characteristic functions is familiar; this theory should be developed as a special case of the multi-dimensional case.
References can be in English, German or French. In fact, the German work "Stochastik: Theorie und Anwendungen" by Meintrup and Schäffler is very much what I have in mind, except their account of the continuity and uniqueness theorems is not self-contained.
Jochen Wengenroth's "Wahrscheinlichkeitstheorie" (de Gruyter, 2008), Theorem 5.6, p. 90 (continuity & uniqueness). Concerning uniqueness, see also the paragraph on p. 84 immediately following the proof of the Portmanteau theorem (theorem 5.2, p. 83). In order to be able to follow theorem 5.6's proof, one is advised to read the entire chapter 5 up to the proof. The writing is concise but well written, and all proofs are given completely.
Two other sources that I've considered, but have not actually read, are:
Karl Stromberg's "Probability for Analysts" (Chapman & Hall, 1994), Theorem 1.11, p. 11 (uniqueness) and Theorem 2.7, p. 23 (continuity).
K. R. Parathasarathy's "Introduction to Probability and Measure" (Hindustan Book Agency, 2005), Proposition 7.3.9, p. 298 (uniqueness) and Proposition 7.3.17, p. 302 (continuity). This book delegates some of the work to exercises.