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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.

cmn1
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  • Do Hamel bases count as sets encountered in general analysis? What about Vitali sets or Bernstein sets? – Asaf Karagila Aug 25 '15 at 00:55
  • @AsafKaragila I would like to think so. Well, I have seen that understanding the construction of Vitali sets is important for clarifying the use of Caratheodory's extension theorem in a measure-theoretic light. Unfortunately, I'm not so sure about Bernstein sets in functional analysis, but I see that their not having the Baire property makes them special at least in the context of understanding the topology of the underlying space. That feels related to my 2nd question If i understand correctly, so thank you for those questions! – cmn1 Aug 25 '15 at 03:54
  • Well, none of the sets I mentioned is Borel (except a Hamel basis for $\Bbb R^n$ over $\Bbb R$, or something like that). – Asaf Karagila Aug 25 '15 at 05:56
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    Kechris states in his book that there are no interesting "natural" examples of Borel sets occuring in analysis or topology beyond the fifth level (i.e., in $\boldsymbol\Sigma_5^0$ or $\boldsymbol\Delta_5^0$). Also, projective sets beyond the second level might not have any regularity property (measurability, Baire-, perfect set-). A very condensed account on this last bound appears in the introduction to Moschovakis' Descriptive Set Theory. – Pedro Sánchez Terraf Aug 25 '15 at 17:51
  • It would be interesting to find a reference for the relationship between nonlinear PDE and descriptive set theory. I imagine some of the non-regularity properties that you mentioned are encountered all over the place in that setting, since I've heard that that field of analysis is littered with pathological sets/ spaces. – cmn1 Nov 06 '15 at 18:46

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There is a theorem of Mazurkiewicz which states that the set $D$ of all differentiable functions in $C([0,1])$ is $\Pi^1_1$-complete. (See Kechris book 33.9 for a proof).

I guess if you want more complicated sets, just take a continuous linear operator $F$ with the appropriate norm and send the set $D$ above by $F$ and you will get a $\Sigma^1_2$ set. (say taking some $F:C([0,1])\to C([0,1])$).

I believe there is a whole industry of results like the above.

Rachid Atmai
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  • Thank you for the reference. Would you know whether there is an online copy of the result/ Kechris text? – cmn1 Nov 06 '15 at 18:43