Signed Measure Examples
Why
Let us consider examples of signed measures.
Examples
Consider an integrable function defined on some measurable space. The extended-real-valued funciton which assigns to each distinguished set the value of the integrating the function over that set is a signed measure.
Suppose $(X, \mathcal{A} , \mu )$ a measure
space and $f: X \to \R $ is an integrable
function.
Define $\nu : \mathcal{A} \to R$ by
\[ \nu (A) = \int_{A} f d\mu . \]
Then $\nu $ is a signed measure.