Is measure theory the study of functions of subsets? My understanding of some of the different sub-fields of analysis is as follows:

*

*Real analysis is the study of functions that map elements of $\mathbb{R}^n$ to elements of $\mathbb{R}^n$.

*Functional analysis is the study of functionals that map elements of some set of functions to elements of set of numbers/vectors, such as $\mathbb{R}^n$, or elements of another set of functions.

*Measure theory is the study of functions that map subsets of some set of numbers/vectors, such as $\mathbb{R}^n$, to elements of another set of numbers/vectors, such as $\mathbb{R}^n$. For example, the probability measure $\mathbb{P}$ maps subsets of a sample space $\Omega$ to $[0,1]$.

Is my understanding correct?
 A: I would respectfully disagree with your last bullet point. You really can not compare real and functional analysis to measure theory, because in measure theory you do not really (you can, but this is rather advanced) care about continuity and differentiability (there are Radon-Nikodym-derivatives but those are not derivatives in the usual sense) of measures. Maybe, you are concerned with limit of measures, but this is again a real and functional analysis topic. There is a reason for why measure theory is not called measure analysis.
So what does it deal with? In measure theory, the sets that are measured are probably more important than the measures itself. It starts with the fact, that the most important measure, the Lebesgue measure, can not measure every set.
Measure theory aims to find a formalization for the term "volume". Some measure theory theorems like Cavalieri's principle are thus very intuitive.
Measures are especially important when defining integrals. They are also an important instrument in functional analysis, for example when it comes to representation of $C^0$-functionals.
(I am sorry for very often repeating the word "measure" but do I have any choice?)
A: It is not correct.
There are many functions from $\mathbb{R}^n$ to $\mathbb{R}^n$ that aren't of much interest to anyone working in analysis (or to anyone at all, really). Saying that it is a study of differentiable functions from $\mathbb{R}^n$ to $\mathbb{R}^n$ would be a much better description, altough still debatable.
Functional analysis does give a lot of attention to function spaces, but certainly is not limited to them. It often considers general (in particular, infinitely dimensional) Hilbert spaces or even topological vector spaces.
Finally, I think it is wrong to consider measure theory as a subfield of analysis. It is a separate field that started out of need for formalisation of the concept of surfeace area and volume and for tools to study those concepts. It is true that in practice, it is very much connected to analysis, but theoretically, it could still function completely separately. Now, regarding your understanding of it, this description is too limited, but also not enlightening at all. First thing, the subsets to which we assign numerical values do not need to be subsets of vector or "number" spaces, whatever you would mean by a number space. Measure theory studies functions that map subsets of some given sets to numerical values in a way following certain rules which are supposed to model our intuitive idea of volume. I believe you should always keep that in mind while thinking what measure theory is about and also while studying it.
