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Lemma: Addition and Scalar Multiplication of Riemann Upper and Lower Integrals

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:

a)

$$\int_a^{b~*}(f+g)(x)dx\le \int_a^{b~*}f(x)dx+\int_a^{b~*}g(x)dx$$

b)

$$\int_{a~*}^{b}(f+g)(x)dx\ge \int_{a~*}^{b}f(x)dx+\int_{a~*}^{b}g(x)dx$$

c)

$$\int_a^{b~*}(\lambda\cdot f)(x)dx=\lambda\cdot\int_a^{b~*}f(x)dx\quad\quad\text{for all }\lambda\ge 0$$

d)

$$\int_{a~*}^{b}(\lambda\cdot f)(x)dx=\lambda\cdot\int_{a~*}^{b}f(x)dx\quad\quad\text{for all }\lambda < 0$$

e)

$$\int_{a~*}^{b}(\lambda\cdot f)(x)dx=\lambda\cdot\int_{a}^{b~*}f(x)dx\quad\quad\text{for all }\lambda < 0$$

| | | | | created: 2016-03-06 15:24:35 | modified: 2020-01-09 05:38:25 | by: bookofproofs | references: [581]

1.Proof: (related to "Addition and Scalar Multiplication of Riemann Upper and Lower Integrals")

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Bibliography (further reading)

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