Let \(f,g:[a,b]\mapsto\mathbb R\) be bounded functions on the closed interval \([a,b]\). The Riemann upper and lower integrals of \(f\) and \(g\) fulfill the following rules:
$$\int_a^{b~*}(f+g)(x)dx\le \int_a^{b~*}f(x)dx+\int_a^{b~*}g(x)dx$$
$$\int_{a~*}^{b}(f+g)(x)dx\ge \int_{a~*}^{b}f(x)dx+\int_{a~*}^{b}g(x)dx$$
$$\int_a^{b~*}(\lambda\cdot f)(x)dx=\lambda\cdot\int_a^{b~*}f(x)dx\quad\quad\text{for all }\lambda\ge 0$$
$$\int_{a~*}^{b}(\lambda\cdot f)(x)dx=\lambda\cdot\int_{a~*}^{b}f(x)dx\quad\quad\text{for all }\lambda < 0$$
$$\int_{a~*}^{b}(\lambda\cdot f)(x)dx=\lambda\cdot\int_{a}^{b~*}f(x)dx\quad\quad\text{for all }\lambda < 0$$
Proofs: 1
Proofs: 1
