Definition (Mutually Singular)

Two Measures $\mu_{1},\mu_{2}$ on $(X,\mathscr{M})$ are called mutually singular if and only if $\exists A_{1},A_{2}\in\mathscr{M}$, $A_{1}\cap A_{2}=\emptyset$ such that $\mu_{i}$ is Concentrated on $A_{i}$, $i=1,2$. We denote this using $$\mu_{1}\perp \mu_{2}$$