Disjoint supports make two measures mutually singular

Let $(\mathbb{R},M,\mu)$ denote the real line with Lebesgue measure $\mu$. Define $\nu_1(A)=\int_Af(x)dx$ where $f(x)=e^{-x}$ for $x>1$, $0$ otherwise, and $\nu_2(A)=\int_Ag(x)dx$ where $g(x)=e^{-1}$ for $0\le x\le1$, $0$ otherwise. Then: