Let $X=[0,1]^n$ endowed with the Euclidean norm and $\mathcal B$ the Borel $\sigma$-algebra on $X$. Let $\lambda$ be the Lebesgue measure and $\mu$ be a finite measure on $(X, \mathcal B)$ with full support (following wikipedia's definition).
Let $\nu_{ac}$ and $\nu_{s}$ be such that $\mu=\nu_{ac}+\nu_{s}$, where $\nu_{ac}$ is absolutely continuous and $\nu_{s}$ and $\lambda$ are mutually singular.
Question 1: Does $\nu_{ac}$ have full support? Is it possible for its support to have Lebesgue measure zero?
Question 2: Are there known results ensuring that $\nu_{ac}$ has full support?
Say $r_n$ is an enumeration of a dense subset of $[0,1]$. Let $$\nu=\sum_n2^{-n}\delta_{r_n}.$$Then $\nu$ has full support and $\nu_{ac}=0$.