Density level sets separate a singular measure ($\nu (Y_n\le \epsilon )\le \epsilon$)

Suppose a finite nonnegative measure has finite-level densities $Y_n$ with respect to a probability measure $\lambda$. The event $\{Y_n\le \epsilon \}$ belongs to that level's sigma-algebra, so its measure is $\int 1_{\{Y_n\le \epsilon \}}{Y}_{n}d\lambda \le \epsilon$. The set where nonnegative $Y_n$ tends to zero is contained in the eventual occurrence of every such level event; it therefore has measure at most $\epsilon$ for every $\epsilon >0$, hence zero. If the same limit set has full $\lambda$-measure, the two measures are singular.