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Riemann linearity

Linearity of the Riemann integral with respect to addition and scalar multiplication.
Riemann linearity

Riemann linearity: Let f,g:[a,b]→Rf,g:[a,b]\to\mathbb R be on the [a,b][a,b], and let α,β∈R\alpha,\beta\in\mathbb R. Then αf+βg\alpha f+\beta g is Riemann integrable on [a,b][a,b], and

∫ab(αf+βg)=α∫abf+β∫abg. \int_a^b (\alpha f+\beta g)=\alpha\int_a^b f+\beta\int_a^b g.

This is the basic linear structure behind the and is used throughout the algebraic manipulation of integrals.