Re: Range of the averages convex?

In article <iY9Wg.1802$ji5.603@trnddc04>,
"johnson" <johnson246@xxxxxxxxxxx> wrote:

"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.

If f is real valued, let a = ess inf f, b = ess sup f. Let eps >
0. By the Lebesgue differentiation theorem, we can find intervals
I and J, of equal length h, such that 1/m(I)*int_I f < a + eps,
1/m(J)*int_J f > b - eps. The function x -> (1/h)int_(x,x+h) f is
continuous, hence its range is an interval, which must contain
[a+eps, b-eps]. This shows S contains (a,b), and since S is
contained in [a,b], S is an interval, hence is convex.
.

Relevant Pages

  • Re: Range of the averages convex?
    ... where \mu is the Lebesgue measure and E are Lebesgue measurable subsets ... Must S be convex? ... David C. Ullrich ...
    (sci.math)
  • Re: Range of the averages convex?
    ... where \mu is the Lebesgue measure and E are Lebesgue measurable subsets ... Must S be convex? ... Now let eps> 0. ... If S has an interior point it's not hard ...
    (sci.math)
  • Re: Range of the averages convex?
    ... where \mu is the Lebesgue measure and E are Lebesgue measurable subsets ... Must S be convex? ... Now let eps> 0. ... If S has an interior point it's not hard ...
    (sci.math)
  • Re: Range of the averages convex?
    ... johnson wrote: ... thing as saying that S in convex. ... Jose Carlos Santos ...
    (sci.math)
  • Re: Range of the averages convex?
    ... Must S be convex? ... If f is real valued, let a = ess inf f, b = ess sup f. ... Let eps> ... By the Lebesgue differentiation theorem, ...
    (sci.math)