Integral of Integrable Function is Additive/Lemma

Lemma

Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $f_1, f_2, g_1, g_2 : X \to \R$ be positive $\mu$-integrable functions with:

$f_1 - f_2 = g_1 - g_2$

Then:

$\ds \int f_1 \rd \mu - \int f_2 \rd \mu = \int g_1 \rd \mu - \int g_2 \rd \mu$

Proof

We have that:

$f_1 + g_2 = g_1 + f_2$

From Pointwise Sum of Measurable Functions is Measurable, we have that:

$f_1 + g_2$ and $g_1 + f_2$ are $\Sigma$-measurable.

So we have:

$\ds \int \paren {f_1 + g_2} \rd \mu = \int \paren {g_1 + f_2} \rd \mu$

So:

$\ds \int f_1 \rd \mu + \int g_2 \rd \mu = \int g_1 \rd \mu + \int f_2 \rd \mu$

from Integral of Positive Measurable Function is Additive.

Since $f_1, f_2, g_1, g_2$ are all $\mu$-integrable, each term is finite, and so:

$\ds \int f_1 \rd \mu - \int f_2 \rd \mu = \int g_1 \rd \mu - \int g_2 \rd \mu$

$\blacksquare$