Let be a -periodic function. It may or may not have a representation as a trigonometric series

A natural question to ask is whether or not the representation of as a trigonometric series is unique, if it has one. It was the consideration of this question that led Cantor to the invention of set theory.

There is a nice writeup of this story in the first part of this article by Alexander Kechris. I’ll give part of the story below.