Re: Range of the averages convex?


"johnson" <johnson246@xxxxxxxxxxx> wrote in message
news:hq9Wg.2204$e65.413@xxxxxxxxxxx

"johnson" <johnson246@xxxxxxxxxxx> wrote in message
news:sm9Wg.2203$e65.1080@xxxxxxxxxxx
Let f be an essentially bounded Lebesgue measurable function on [0,1].
Consider the set
\int_E f d\mu
where \mu is the Lebesgue measure and E are Lebesgue measurable subsets
of [0,1].
Must f be convex?

Correction:

Let f be an essentially bounded Lebesgue measurable function on [0,1].
Consider the set S=
\int_E f d\mu
where \mu is the Lebesgue measure and E are Lebesgue measurable subsets of
[0,1].
Must S be convex?


Sorry. I meant 1/mu(E) times \int_E f d\mu
for E such that mu(E)>0.


.