My question comes from a paper and I have no concrete background on this.
Let $\mu$ be an probability measure over $(X,M(X))$, where $M(X)$ is the Borel $\sigma$-algebra over $X.$ So $\mu$ is nonnegative. $f(x): X\longrightarrow \mathbb{R}$, excluding $\pm\infty$.
Consider the following: $$\int_X f(x) d\mu(x) = \int_X d\big(\nu^+(x)-\nu^-(x)\big) =\int_X d\nu^+(x)-\int_Xd\nu^-(x)$$
where $\nu^+$, $\nu^-\in\mathcal{M}(X)$ are nonnegative measures. $\mathcal{M}(X)$ is a vector space of measures on $X$.
LHS is just a normal integration; however, RHS becomes the algebraic computation of unsigned measures. It seems to me that we can create a signed measure $\nu^+-\nu^-$ by including $f(x)$, which is a real number, into $d\mu(x)$.
My questions are
- Now $\nu^+$, $\nu^-$ are not probability measure and not nonnegative but are still measures?
- This new measure encodes the information of $f(x)$?
- Are there any other examples or applications of such technique?
If $\mu$ is a measure (non-negative, but not necessarily finite), and $f \in L^1(\mu)$, then $\nu(A) := \int_A f \, d\mu$ defines a signed measure. By the Hahn decomposition theorem this can be decomposed into two non-negative mutually singular measures in a unique way, $\nu = \nu^+ - \nu^-$.
If $\mu$ is a probability measure, i.e. $\mu(X)=1,$ then $\nu^\pm$ are not necessarily probability measures.
$\nu$ or $\{\nu^+, \nu^-\}$ encodes the information of $f$ almost everywhere. If $f$ changes on a $\mu$-null set, then $\nu$ will not change.