User:Caliburn/s/mt/Radon-Nikodym Theorem/Signed Measure
Theorem
Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ be a $\sigma$-finite measure on $\struct {X, \Sigma}$.
Let $\nu$ be a finite signed measure on $\struct {X, \Sigma}$ such that:
- $\nu$ is absolutely continuous with respect to $\mu$.
Then there exists a $\Sigma$-measurable function $h : X \to \hointr 0 \infty$ such that:
- $\ds \map \nu A = \int_A h \rd \mu$
for each $A \in \Sigma$.
Further, if $h_1 : X \to \hointr 0 \infty$ and $h_2 : X \to \hointr 0 \infty$ are such that:
- $\ds \map \nu A = \int_A h_1 \rd \mu = \int_A h_2 \rd \mu$
for each $A \in \Sigma$, then:
- $h_1 = h_2$ $\mu$-almost everywhere.
Proof
Existence
Let $\tuple {\nu^+, \nu^-}$ be the Jordan decomposition of $\nu$.
From Absolute Continuity of Signed Measure in terms of Jordan Decomposition, we have:
- $\nu^+$ and $\nu^-$ are absolutely continuous with respect to $\mu$.
From the Radon-Nikodym Theorem, there exists $\Sigma$-measurable function $g_1 : X \to \hointr 0 \infty$ such that:
- $\ds \map {\nu^+} A = \int_A g_1 \rd \mu$
for each $A \in \Sigma$.
Applying the Radon-Nikodym Theorem again, there exists $\Sigma$-measurable function $g_2 : X \to \hointr 0 \infty$ such that:
- $\ds \map {\nu^-} A = \int_A g_2 \rd \mu$
for each $A \in \Sigma$.
From Jordan Decomposition of Finite Signed Measure, we have:
- $\nu^+$ and $\nu^-$ are finite measures.
So, we have:
- $\ds \map {\nu^+} X = \int g_1 \rd \mu < \infty$
and:
- $\ds \map {\nu^-} X = \int g_2 \rd \mu < \infty$
so $g_1$ and $g_2$ are $\mu$-integrable functions.
Then, for each $A \in \Sigma$, we have:
\(\ds \map \nu A\) | \(=\) | \(\ds \map {\nu^+} A - \map {\nu^-} A\) | Definition of Jordan Decomposition | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_A g_1 \rd \mu - \int_A g_2 \rd \mu\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int_A \paren {g_1 - g_2} \rd \mu\) |
Essential Uniqueness
Suppose that $\Sigma$-measurable $h_1 : X \to \hointr 0 \infty$ and $h_2 : X \to \hointr 0 \infty$ are such that:
- $\ds \map \nu A = \int_A h_1 \rd \mu = \int_A h_2 \rd \mu$
for each $A \in \Sigma$.
Let $\tuple {P, N}$ be a Hahn decomposition of $\mu$.
From the definition of the Jordan decomposition and Uniqueness of Jordan Decomposition, we have:
- $\map {\nu^+} A = \map \nu {A \cap P}$
and:
- $\map {\nu^-} A = -\map \nu {A \cap N}$
for each $A \in \Sigma$.
We therefore have:
\(\ds \map {\nu^+} A\) | \(=\) | \(\ds \int_{P \cap A} h_1 \rd \mu\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \paren {h_1 \times \chi_{P \cap A} } \rd \mu\) | Definition of Integral of Integrable Function over Measurable Set | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \paren {h_1 \times \chi_P \chi_A} \rd \mu\) | Characteristic Function of Intersection | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_A \paren {h_1 \times \chi_P} \rd \mu\) | Definition of Integral of Integrable Function over Measurable Set |
and:
\(\ds \map {\nu^+} A\) | \(=\) | \(\ds \int_{P \cap A} h_2 \rd \mu\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \paren {h_2 \times \chi_{P \cap A} } \rd \mu\) | Definition of Integral of Integrable Function over Measurable Set | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \paren {h_2 \times \chi_P \chi_A} \rd \mu\) | Characteristic Function of Intersection | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_A \paren {h_2 \times \chi_P} \rd \mu\) | Definition of Integral of Integrable Function over Measurable Set |
So $h_1 \times \chi_P : X \to \hointl 0 \infty$ and $h_2 \times \chi_P : X \to \hointl 0 \infty$ are such that:
- $\ds \map {\nu^+} A = \int_A \paren {h_1 \times \chi_P} \rd \mu = \int_A \paren {h_2 \times \chi_P} \rd \mu$
From the essential uniqueness part of the Radon-Nikodym Theorem, we therefore have:
- $h_1 \times \chi_P = h_2 \times \chi_P$ $\mu$-almost everywhere.
Similarly, we have:
\(\ds \map {\nu^-} A\) | \(=\) | \(\ds -\int_{N \cap A} h_1 \rd \mu\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\int \paren {h_1 \times \chi_{N \cap A} } \rd \mu\) | Definition of Integral of Integrable Function over Measurable Set | |||||||||||
\(\ds \) | \(=\) | \(\ds -\int \paren {h_1 \times \chi_N \chi_A} \rd \mu\) | Characteristic Function of Intersection | |||||||||||
\(\ds \) | \(=\) | \(\ds -\int_A \paren {h_1 \times \chi_N} \rd \mu\) | Definition of Integral of Integrable Function over Measurable Set |
and:
\(\ds \map {\nu^-} A\) | \(=\) | \(\ds -\int_{N \cap A} h_2 \rd \mu\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\int \paren {h_2 \times \chi_{N \cap A} } \rd \mu\) | Definition of Integral of Integrable Function over Measurable Set | |||||||||||
\(\ds \) | \(=\) | \(\ds -\int \paren {h_2 \times \chi_N \chi_A} \rd \mu\) | Characteristic Function of Intersection | |||||||||||
\(\ds \) | \(=\) | \(\ds -\int_A \paren {h_2 \times \chi_N} \rd \mu\) | Definition of Integral of Integrable Function over Measurable Set |
So $h_1 \times \chi_N : X \to \hointl 0 \infty$ and $h_2 \times \chi_N : X \to \hointl 0 \infty$ are such that:
- $\ds \map {\nu^-} A = \int_A \paren {h_1 \times \chi_N} \rd \mu = \int_A \paren {h_2 \times \chi_N} \rd \mu$
From the essential uniqueness part of the Radon-Nikodym Theorem, we therefore have:
- $h_1 \times \chi_N = h_2 \times \chi_N$ $\mu$-almost everywhere.
From the definition of Hahn decomposition, we have that $P$ and $N$ are disjoint with:
- $X = P \cup N$
so that:
\(\ds h_1\) | \(=\) | \(\ds h_1 \chi_{P \cup N}\) | Characteristic Function of Universe | |||||||||||
\(\ds \) | \(=\) | \(\ds h_1 \paren {\chi_P + \chi_N}\) | Characteristic Function of Disjoint Union: Corollary | |||||||||||
\(\ds \) | \(=\) | \(\ds h_1 \times \chi_P + h_1 \times \chi_N\) |
and:
\(\ds h_2\) | \(=\) | \(\ds h_2 \chi_{P \cup N}\) | Characteristic Function of Universe | |||||||||||
\(\ds \) | \(=\) | \(\ds h_2 \paren {\chi_P + \chi_N}\) | Characteristic Function of Disjoint Union: Corollary | |||||||||||
\(\ds \) | \(=\) | \(\ds h_2 \times \chi_P + h_2 \times \chi_N\) |
From Pointwise Addition preserves A.E. Equality: General Result, we have:
- $h_1 \times \chi_P + h_1 \times \chi_N = h_2 \times \chi_P + h_2 \times \chi_N$ $\mu$-almost everywhere
so that:
- $h_1 = h_2$ $\mu$-almost everywhere.