MathDB

Let

C

denote the set of functions

f(x)

, integrable (according to either Riemann or Lebesgue) on

(a,b)

, with

0\le f(x)\le1

. An element

\phi(x)\in C

is said to be an "extreme point" of

C

if it can not be represented as the arithmetical mean of two different elements of

C

. Find the extreme points of

C

and the functions

f(x)\in C

which can be obtained as "weak limits" of extreme points

\phi_n(x)

C

. (The latter means that \lim_{n\to \infty}\int_a^b \phi_n(x)h(x)\,dx\equal{}\int_a^bf(x)h(x)\,dx holds for every integrable function

h(x)

Problem Statement