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Topic: Averages of a Function
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Maury Barbato

Posts: 791
From: University Federico II of Naples
Registered: 3/15/05
Averages of a Function
Posted: Jul 5, 2013 10:58 AM
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Hello,
let m be the Lebesgue measure on R, and f:R -> C a
complex-valued 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!



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