Integral of Integrable Function is Additive/Corollary 2

Corollary
Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $f, g: X \to \overline \R$ be $\mu$-integrable functions.

Suppose that the pointwise difference $f - g$ is well-defined.

Then $f - g$ is $\mu$-integrable, with:


 * $\displaystyle \int \paren {f - g} \rd \mu = \int f \rd \mu - \int g \rd \mu$

Proof
From Integral of Integrable Function is Homogeneous, we have that:


 * $-g$ is $\mu$-integrable.

From Integral of Integrable Function is Additive: Corollary 1, we therefore have:


 * $f - g$ is $\mu$-integrable

with:


 * $\ds \int \paren {f - g} \rd \mu = \int f \rd \mu + \int \paren {-g} \rd \mu$

Then, from Integral of Integrable Function is Homogeneous, we have:


 * $\ds \int \paren {-g} \rd \mu = -\int g \rd \mu$

so:


 * $\ds \int \paren {f - g} \rd \mu = \int f \rd \mu - \int g \rd \mu$