In analysis, it is common to encounter subsets of $\mathbb R$ (or even $\mathbb R^n$) which appear to be "well-behaved", especially with regard to properties like being measurable, compactness, etc. It is core to descriptive set theory (DST) that one is able to impose a classification of such subsets by closure properties of varying degree. Loosely speaking, for example, one obtains the Borel hierarchy by applying a certain transfinite induction on open/closed sets indexed by ordinal numbers $\alpha_k < \omega_1$, taking unions at limit stages, and so on. On the face of it, it appears that the complexity of the sets used in real analysis and measure theory are of low complexity in the Borel hierarchy, and sets in functional analysis (I am thinking spaces of functions, etc) are of a strictly higher complexity, but I am unsure where or how their relative complexity is classified in the known hierarchies.
My main question can be split into some sub-questions:
1) What are the most "complicated" sets which one encounters in general analysis, and how may one translate complexity classification in DST to concrete problems in real analysis?
2) Is there any reasonable method in DST to measure the set-theoretic complexity of general spaces? (Here I am thinking along the lines of $L^p$ and Sobolev spaces).
I apologize if (2) is a silly question, but a nice reference noting the recent applications of DST to functional analysis is found here http://www.math.uiuc.edu/~anush/Notes/dst_lectures.pdf in the introduction, namely in connection with a background for classification problems in a concrete setting. More references would be appreciated.