Prove the linearity property of integrals of step functions

by RoRi

July 20, 2015

Prove that for step function $s(x), t(x)$ defined on an interval $[a,b]$ and for constants $c_1, c_2 \in \mathbb{R}$ , we have

$\int_a^b \left(c_1 s(x) + c_2 t(x) \right) \, dx = c_1 \int_a^b s(x) \, dx + c_2 \int_a^b t(x) \, dx.$

In Apostol, this is Theorem 1.4.

Proof. By the additive property of the integral of step functions we know

$\int_a^b ( c_1 s(x) + c_2 t(x) ) \, dx = \int_a^b c_1 s(x) \, dx + \int_a^b c_2 t(x) \, dx.$

Then, by the homogeneous property (Theorem 1.3 in Apostol) we know

$\int_a^b c_1 s(x) \, dx + \int_a^b c_2 t(x) \, dx = c_1 \int_a^b s(x) \, dx + c_2 \int_a^b t(x) \, dx. \qquad \blacksquare$

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