Integral of Bounded Measurable Function with respect to Finite Signed Measure is Well-Defined

Theorem

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

Let $f : X \to \R$ be a bounded $\Sigma$-measurable function.

Let $\mu$ be a finite signed measure on $\struct {X, \Sigma}$.

Let $\tuple {\mu^+, \mu^-}$ be the Jordan decomposition of $\mu$.

Then the $\mu$-integral of $f$ defined by:

$\ds \int f \rd \mu = \int f \rd \mu^+ - \int f \rd \mu^-$

is well-defined.

Proof

We show that $f$ is $\mu^+$-integrable and $\mu^-$-integrable.

We will then have:

$\ds -\infty < \int f \rd \mu^+ < \infty$

and:

$\ds -\infty < \int f \rd \mu^- < \infty$

So that:

$\ds \int f \rd \mu^+ - \int f \rd \mu^-$

is well-defined.

Since $f$ is bounded, there exists $M > 0$ such that:

$\size {\map f x} \le M$

for each $x \in X$.

From Jordan Decomposition of Finite Signed Measure, we have:

$\mu^+$ and $\mu^-$ are finite.

That is:

$\map {\mu^+} X < \infty$

and:

$\map {\mu^-} X < \infty$

We therefore have:

\(\ds \int \size f \rd \mu^+\)

\(\le\)

\(\ds \int M \rd \mu^+\)

Measure is Monotone

\(\ds \)

\(=\)

\(\ds M \int \rd \mu^+\)

Integral of Positive Measurable Function is Positive Homogeneous

\(\ds \)

\(=\)

\(\ds M \map {\mu^+} X\)

Integral of Characteristic Function: Corollary

\(\ds \)

\(<\)

\(\ds \infty\)

so:

\(\ds \int \size f \rd \mu^-\)

\(\le\)

\(\ds \int M \rd \mu^-\)

Measure is Monotone

\(\ds \)

\(=\)

\(\ds M \int \rd \mu^-\)

Integral of Positive Measurable Function is Positive Homogeneous

\(\ds \)

\(=\)

\(\ds M \map {\mu^-} X\)

Integral of Characteristic Function: Corollary

\(\ds \)

\(<\)

\(\ds \infty\)

So:

$\size f$ is $\mu^+$-integrable

and:

$\size f$ is $\mu^-$-integrable.

From Characterization of Integrable Functions, we have:

$f$ is $\mu^+$-integrable and $\mu^-$-integrable

as required.

$\blacksquare$