Linearity and Monotony of the Riemann Integral for Step Functions

Proof (related to "Linearity and Monotony of the Riemann Integral for Step Functions")

Let \(\phi,\psi:[a,b]\mapsto\mathbb R\) be step functions over the closed interval \([a,b]\). According to the definition of the Riemann integral for step functions, both integrals
\[\int_a^b\phi(x)dx=\sum_{i=1}^n\phi(x_i)(x_i-x_{i-1})\quad\quad\text{and}\quad\quad\int_a^b\psi(x)dx=\sum_{i=1}^n\psi(x_i)(x_i-x_{i-1})\]
can be calculated without loss of generality using the same partition \(a=x_0 < x_1 < \ldots < x_{n-1} < x_n=b\). Therefore, the stated

linearity rules:

\[\int_a^b(\phi+\psi)(x)dx=\int_a^b\phi(x)dx+\int_a^b\psi(x)dx\]
\[\int_a^b(\lambda\cdot \phi)(x)dx=\lambda\cdot\int_a^b\phi(x)dx\quad\quad(\text{for all }\lambda\in\mathbb R)\]

and the monotony rule:

\[\phi\le \psi\Rightarrow \int_a^b\phi(x)dx\le \int_a^b\psi(x)dx\]

are trivial. In the monotony rule, “\(\phi\le \psi\)” means the order relation for step functions.

References

[581] Forster Otto: “Analysis 1, Differential- und Integralrechnung einer Veränderlichen”, Vieweg Studium, 1983

Discussion