Jordan Decomposition

This post is inspired by Professor Tom Fischer’s writeup.

Recall the Hahn decomposition theorem:

Theorem 1. If $\mu$ is a signed measure on $(\Omega, \mathcal F)$ , then there exist disjoint measurable subsets $P, N \in \mathcal F$ such that

$\mu (\cdot \cap P) \in [0, \infty),\quad \mu (\cdot \cap N) \in (-\infty, 0] ,\quad P \sqcup N = \Omega.$

We call $P$ a positive set, denoted $P \geq 0$ , and $N$ a negative set, denoted $N \leq 0$ , and we call the pair $(P, Q)$ a Hahn decomposition for $\mu$ .

Call a set $M$ is $\mu$ –null if it is positive and negative: $\mu( \cdot \cap M) \in \{0\}$ .

Define the symmetric difference by

$K\triangle L := (K \backslash L) \sqcup (L \backslash K).$

We leave it as an exercise to verify that

$K \cup L = (K \cap L) \sqcup (K\triangle L).$

Problem 1. Let $\mu$ be a signed measure on $(\Omega, \mathcal F)$ and $(P_1,N_1)$ , $(P_2,N_2)$ be two Hahn decompositions for $\mu$ . Show that $P_1 \triangle P_2$ and $N_1 \triangle N_2$ are $\mu$ -null.

(Click for Solution)

Solution. Fix $K \in \mathcal F$ . Then

$\begin{aligned} \mu(K \cap (P_1 \triangle P_2)) &= \mu(K \cap P_1 \backslash P_2) + \mu(K \cap P_2 \backslash P_1) \\ &= \mu(K \cap P_1 \cap N_2) + \mu(K \cap P_2 \cap N_2) \\ &= \mu(K \cap P_1 \cap N_2) + \mu(\emptyset) \\ &= \mu(K \cap P_1 \cap N_2). \end{aligned}$

Using Theorem 1,

$\begin{aligned} 0 &\leq \mu((K \cap N_2) \cap P_1) \\ &= \mu(K \cap P_1 \cap N_2) \\ &= \mu((K \cap P_1) \cap N_2) \leq 0, \end{aligned}$

so $\mu(K \cap (P_1 \triangle P_2)) = \mu(K \cap P_1 \cap N_2) = 0$ , as required.

We now state the Jordan decomposition theorem.

Problem 2. Let $\mu$ be a finite signed measure on $(\Omega, \mathcal F)$ . Construct unique finite measures $\mu^+, \mu^-$ on $(\Omega , \mathcal F)$ such that:

$\mu = \mu^+ - \mu^-$ , and
for any Hahn decomposition $(P, N)$ of $(\Omega, \mathcal F)$ , $\mu^+(K) = 0$ for $K \subseteq N$ and $\mu^-(K) = 0$ for $K \subseteq P$ .

We call $(\mu^+, \mu^-)$ the (unique) Jordan decomposition of $\mu$ .

(Click for Solution)

Solution. Use Theorem 1 to construct a Hahn decomposition $(P_1, N_1)$ . Define

$\begin{aligned} \mu^+ &:= \mu(\cdot \cap P_1) \in [0, \infty),\\ \mu^- &:= -\mu(\cdot \cap N_1) \in [0, \infty). \end{aligned}$

Then for any $K \in \mathcal F$ ,

$\begin{aligned} \mu(K) &= \mu(K \cap P_1) + \mu(K \cap N_1) \\ &= \mu^+(K) - \mu^-(K) \\ &= (\mu^+ - \mu^-)(K). \end{aligned}$

Hence, $\mu = \mu^+ - \mu^-$ .

Now fix any Hahn decomposition $(P_2, N_2)$ . Then

$\begin{aligned} P_1 &= P_1 \cap (P_1 \cup P_2) \\ &= P_1 \cap ((P_1 \cap P_2) \sqcup (P_1 \triangle P_2)) \\ &= (P_1 \cap (P_1 \cap P_2)) \sqcup (P_1 \cap (P_1 \triangle P_2)) \\ &= (P_1 \cap P_2) \sqcup (P_1 \cap (P_1 \triangle P_2)).\end{aligned}$

Fix $K \subseteq N_2$ . Then

$\begin{aligned} \mu^+(K) &= \mu(K \cap P_1) \\ &= \mu(K \cap P_1 \cap P_2) + \mu(K \cap P_1 \cap (P_1 \triangle P_2)) \\ &= \mu(K \cap \emptyset) + \mu((K \cap P_1) \cap (P_1 \triangle P_2)) \\ &= 0 + 0 = 0. \end{aligned}$