Simple Integral Monotonicity
Why
If one rectangle contains another rectangle, the area of the first should be larger than the area of the second. Our definition of integral for simple functions carries this property.
Result
The simple non-negative integral operator is
monotone.
Suppose $(X, \mathcal{A} , \mu )$ is a measure
space.
Suppose that $f, g \in \SimpleF_+(X)$ with $f
\leq g$.
Then $f - g \in \SimpleF_+(X)$, so
\[ \begin{aligned} \int g d\mu &= \int (f + (g - f)) d\mu \\ &\overset{(a)}{=} \int f d\mu + \int (g - f) d\mu \\ &\overset{(b)}{\geq} \int f d\mu \end{aligned} \]
where (a) follows from the linearity and (b) from non-negativity of the non-negative simple integral operator.