Common Ground between Real Analysis and Measure Theory

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!

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What is your definition of "field"? – Jesse Madnick Oct 3 '12 at 23:33

Rather than answer your other questions (they are in some sense just not the kinds of questions one asks in measure theory) let me just make some general comments. Real analysis is in some sense the study of metric spaces. Any metric space gives rise to a topological space, and any topological space gives rise to a measurable space with the same underlying set whose $\sigma$-algebra is the Borel $\sigma$-algebra generated by the open sets. Now one can ask for measures on this $\sigma$-algebra. A fundamental such example is the Lebesgue measure on $\mathbb{R}^n$, which is a very powerful and flexible way to integrate functions on $\mathbb{R}^n$ generalizing the Riemann integral.
Measure theory allows rigorous constructions of a very important class of metric spaces, namely the $L^p$-spaces. The techniques you're currently learning in real analysis will be important for understanding these spaces, which are studied in functional analysis.
@Elements: it is not quite right to say that metric spaces are a type of topological space. They give rise to a type of topological space, but this process is lossy (for example it forgets which sequences are Cauchy). A topological space is a pair consisting of a set $X$ and some collection of subsets of $X$. It gives rise to a measurable space, which again is a pair consisting of a set and some collection of subsets of $X$. The set $X$ is the same, but the subsets are the $\sigma$-algebra generated by the topology, not the topology itself. Again, this process is lossy. – Qiaochu Yuan Oct 3 '12 at 22:03
@QiaochuYuan: So you're saying a $\sigma$-field is not an example of a field of sets? (If I'm understanding correctly, this is the OP's first bullet point.) – Jesse Madnick Oct 3 '12 at 23:31