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Maury Barbato
Posts:
792
From:
University Federico II of Naples
Registered:
3/15/05


Averages of a Function
Posted:
Jul 5, 2013 10:58 AM


Hello, let m be the Lebesgue measure on R, and f:R > C a complexvalued function in L^{\infty}. Let A_f be the set of all averages of f
1/m(E) int_E f(x) dx,
where E ranges over all Lebesgue measurable sets such that 0 < m(E) < \infty. Is A_f convex?
I would be very glad if you had some idea about this problem (which is a particular case of Exercise 3.19 in Rudin, Real and Complex Analysis).
Some related results were proved by prof. David C Ullrich, the World Wide Web and others in the posts
http://mathforum.org/kb/thread.jspa? forumID=253&threadID=559133&messageID=1677532
and
http://mathforum.org/kb/message.jspa?messageID=5228922
Thank you very much for your attention. My Best Regards, Maury Barbato
PS Rudin asks the general question: does there exist a measure space (X, M, \mu) such that for every complex valued measurable function f:X > C a in L^{\infty}(\mu), the set of averages A_f (defined as above with m replaced by \mu) is convex? Obviously, I don't even the answer to this question!



