Measures in Jordan Decomposition of Complex Measure are Finite

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Theorem

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

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

Let $\tuple {\mu_1, \mu_2, \mu_3, \mu_4}$ be the Jordan decomposition of $\mu$.

Then:

$\mu_1$, $\mu_2$, $\mu_3$ and $\mu_4$ are finite.

Proof

Let $\mu_R$ be the real part of $\mu$.

Let $\mu_I$ be the imaginary part of $\mu$.

Then:

$\tuple {\mu_1, \mu_2}$ is the Jordan decomposition of $\mu_R$

and:

$\tuple {\mu_3, \mu_4}$ is the Jordan decomposition of $\mu_I$.

From the definition of the real part and imaginary part, we have that both $\mu_R$ and $\mu_I$ are finite signed measures.

Now applying Jordan Decomposition of Finite Signed Measure to $\mu_R$, we have:

$\mu_1$ and $\mu_2$ are finite.

Applying Jordan Decomposition of Finite Signed Measure to $\mu_I$, we have:

$\mu_4$ and $\mu_4$ are finite

hence the result.

$\blacksquare$