I am working through a proof of the following theorem:
'If $f$ and $g$ are step functions having $f(x) \geq g(x)$ for all real values $x$, then $\int f \geq \int g$.
So far I understand and thus have stated that $(f-g)(x) = f(x) - g(x)$ is a step function. We can rewrite $f-g$ as $\sum_{j=1}^{m}c_j \cdot \chi_{I_j}$. It then states that since $f(x) \geq g(x)$ for all $x$, the co-efficents $c_j$ forming the function values must be positive. But why is that so?
Any help would be greatly appreciated.
While we can write it with a mix of coefficients in which some are negative, there's at least one representation with positive coefficients.
We subdivide the domain based on the values of $f-g$. Write $\mathbb{R}$ as a disjoint union of intervals $I_j$, such that $f-g$ is constant on each $I_j$. These intervals will be intersections of the intervals for $f$ and those for $g$.
Then, since its a disjoint union, $0\le f(x)=c_j$ for any $x\in I_j$.