I have the following linear optimization problem. $$ \max \int_0^1 w(t) dt $$ subject to $$ \int_0^1 w(t) \, x_i(t) \, dt \geq 0, \quad i=1,\dots,n $$ and $$ 0 \leq w(t) \leq 1 \quad \text{for all} \quad t\in[0,1]. $$ Here, $x_1(t), \dots, x_n(t)$ are some given functions $x_i:[0,1]\to{\mathbb R}$, and the optimization is over all measurable functions $w:[0,1]\to{\mathbb R}$ satisfying the constraints.
I would like to prove that there exists a maximiser $w^*(t)$ such that for every $t\in[0,1]$ either $w^*(t)=0$ or $w^*(t)=1$.
This is "obvious" from the intuition that optimal solution for the linear program should be on the "corner" of the feasibility set. However, in this example we have uncountably many variables ($w(t)$ for each $t\in[0,1]$) and uncountably many constraints. Can anyone suggest a good reference for study of such linear programs, and/or provide a direct solution for this problem?