I am currently studying mathematics as an undergrad. Over the last year I discovered mathematical logic and set theory and took some courses in these subjects. Especially set theory is really appealing to me, so naturally I am interested in deepening my knowledge of it. I think set theory is the most interesting part of mathematics I've seen so far. Currently, my aim is to become a professional set theorist. To do so I obviously have to learn more about set theory and take every chance to attend set theory courses. As an undergraduate student, I have (and of course want) to cover different areas of mathematics and not only set theory.
So I wondered - which other areas of mathematics are important in modern set theory or have an influence there? What other areas should one study, if he wants to become professional in set theory?
I am happy to get any general advice on which areas to study, but would also like to hear especially which parts of these areas are important (for example measure theory in analysis).