MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Is it true that all universally measurable sets (say on $[0,1]$ ) have the perfect set property?
I am not an expert in this at all and the answer may be known, but I was not able to find it. I know that all Borel, analytic, and projective sets have the perfect set property and are universally measurable but in such generality the answer to my question may be false.

share|cite|improve this question
1  
Are you working in ZFC? If so, you can take any uncountable universally null set as a counterexample. If it contained a continuous injective image of $2^\omega$, you'd be able to push forward the product measure to contradict its universally nullness (nullity?). – Clinton Conley Mar 28 '12 at 13:14
    
Perhaps the "perfect set property" is: given any nonzero finite measure $\mu$ on the sigma-algebra, there is a perfect set $F$ with $\mu(F)>0$. – Gerald Edgar Mar 28 '12 at 13:29
    
I guess also require $\mu$ to be atomless. – Gerald Edgar Mar 28 '12 at 13:31
    
I assumed perfect set property meant the standard thing, although in retrospect it's somewhat suspicious that projective sets are claimed to possess it. (Also I wish we could edit comments -- I was so thrown by "nullness" vs. "nullity" that I forgot to demote "universally" from adverb to adjective.) – Clinton Conley Mar 28 '12 at 13:52
1  
Perhaps the question's assertion that projective sets are all universally measurable and have the PSP means that Detelin wants to assume projective determinacy. – Ed Dean Mar 28 '12 at 13:57

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.