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I know the construction by Lusin of a measurable set that is non-Borel. But that set turns out to be analytic. Are there some examples of non-analytic sets that are measurable?

Maybe the set that one constructs using the method discussed in Lebesgue measurable but not Borel measurable is such an example?

NPHA
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  • There are plenty of "natural examples" of co-analytic (hence measurable) sets that are not analytic. Several examples can be found in Howard Becker's1992 paper Descriptive set theoretic phenomena in analysis and topology. For something a bit more distant from being an analytic set, perhaps the references and remarks I and others posted here might be of use. – Dave L. Renfro Aug 17 '17 at 14:40
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    I guess you want "positive measure", right? Otherwise just by cardinality arguments, most subsets of the Cantor sets are not analytic (or even projective), and they are all measurable by the virtue of being subsets of a null set. – Asaf Karagila Aug 17 '17 at 15:04
  • @AsafKaragila Well when I posted the question I thought of any set. But the set constructed for example in http://www.math3ma.com/mathema/2015/8/9/lebesgue-but-not-borel will do the trick if Im correct. So a set of positive measure would be more of a question. – NPHA Aug 17 '17 at 15:19
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    @Asaf Karagila: For positive measure, just use the union of your favorite measure zero set and any set of positive measure that is positively separated from the measure zero set, for example pick the measure zero set to be in $[0,1]$ and let the positive measure set be all of $[2,3].$ Less trivial would be a set whose intersection with every nonempty open interval has positive measure and is not analytic (i.e. everywhere of positive measure and nowhere analytic). – Dave L. Renfro Aug 17 '17 at 16:17
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    The complement of an analytic but not Borel set is not analytic. If a set is measurable so is its complement. (In fact, you can show every analytic set is measurable.) – William Aug 17 '17 at 17:49
  • @William That's nice observation! If you post it as answer I can accept it. – NPHA Aug 17 '17 at 19:15

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