Interchangeable Singular Decomposition
Why
Suppose a measure is not interchangeable with another. Then what. We can separate out the troublesome piece; perhaps it can be handled separately.
Result
Let $(X, \mathcal{A} )$ be a measurable space. Let $\mu $ be a measure on $(X, \mathcal{A} )$. Let $\nu $ be a finite signed measure or complex measure or $\sigma $-finite measure on $(X, \mathcal{A} )$. There there is a unique decomposition $\nu = \nu _a + \nu _s$ where $\nu _a \ll \mu $ and $\nu _s \perp \mu $.
The above is also called Lebesgue's Decomposition Theorem