The theorem you are discussing is a basic result in probability/measure theory. The proofs would more likely be found in books on probability or statistical theory. I found the analogous result for characteristic functions given in Hoel Port and Stone pp 205-208. Tucker pp 51-53. and Chung pp 151-155 This is the Third Edition. I have the second ...

Hoel Port Stone Introduction to Probability Theory The Houghtolo MUllin Series in Statistics under the Editorship of H... An Introduction to Probability Theory An introduction to probability theory Christel Geiss and Stefan Geiss February 19, 2004 2 Contents 1 Probability spa...

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