Find a function with integral bounded from below and with prescribed properties

Question

Let $(X,\mathcal A,\mu)$ be a measure space such that $\mu(X)<\infty$. Moreover assume that for any $i=1,\ldots, N$ the function $f_i:X\to [0,+\infty[$ is measurable and such that $$ \int_X f_id\mu>C $$

for a fixed real contant $C$. Is it possible to find a measurable function $f:X\to [0,+\infty[$ satisfying the following properties (at the same time)?

• $f(x)\le f_i(x)$ for all $x\in X$ and any $i=1,\ldots, N$
• $\int_X f d\mu>C$

Answer 1

No. Let $N=2$, $C=1$, $(X, \mathcal{A}, \mu)$ be the probability space $([0,1], \mathscr{B}, \lambda)$, and define $$ f_1:=3\chi_{[0,\frac{1}{2}]} \quad \text{ and }\quad f_2:=f_1:=3\chi_{(\frac{1}{2},1]}. $$ If $f: X\to [0,\infty)$ satifies $f\le f_1$ and $f\le f_2$ then $f=0$. However, it is false that $\int f \mathrm{d}\mu>1$.