I'm currently taking two introductory classes in Real Analysis (Rudin textbook) and Measure Theory (no textbook - but the material we cover is very standard).
It seems as if there is a huge overlap between the material that is covered in both classes. In particular, I believe that Measure Theory is more of a specific application of Real Analysis. That said, I'm having a lot of difficulty seeing how the two fields relate to one another.
This is all very broad, so here are some questions that I have:
Are $\sigma$-fields a subtype of field?
What are the "real analysis" type properties of a Borel set? (i.e. is it closed? open?compact?)"
What are the "real analysis" type properties of a Random Variable?
What are the "real analysis"-type properties of a Measure?
Have I even covered enough material to see the "common ground" between these subjects? (in Real Analysis, we've covered Ch. 1-2 of Rudin and in Measure Theory, we've covered probability spaces and random variables).
Any other insights are very much appreciated!