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:

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

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:

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

$\blacksquare$