Intersection of Positive Set and Negative Set is Null Set

Theorem

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

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

Let $P$ be a $\mu$-positive set.

Let $N$ be a $\mu$-negative set.

Then:

$P \cap N$ is a $\mu$-null set.

Proof

Note that, from Sigma-Algebra Closed under Countable Intersection:

$P \cap N \in \Sigma$

We aim to show that:

for each $E \in \Sigma$ with $E \subseteq P \cap N$ we have $\map \mu E = 0$.

Note first that from Intersection is Subset, we have:

$P \cap N \subseteq P$

so that:

$E \subseteq P$

So, since $P$ is $\mu$-positive, we have:

$\map \mu E \ge 0$

We also have that, from Intersection is Subset:

$P \cap N \subseteq N$

so that:

$E \subseteq N$

Since $N$ is $\mu$-negative, we have:

$\map \mu E \le 0$

Since:

$\map \mu E \ge 0$ and $\map \mu E \le 0$

we have:

$\map \mu E = 0$

So:

for each $E \in \Sigma$ with $E \subseteq P \cap N$ we have $\map \mu E = 0$.

So:

$P \cap N$ is a $\mu$-null set.

$\blacksquare$