= Density level sets separate a singular measure
{title2=$\nu(Y_n\leq\varepsilon)\leq\varepsilon$}
Suppose a finite nonnegative <measure> has finite-level densities $Y_n$ with respect to a <probability> <measure> $\lambda$. The event $\{Y_n\leq\varepsilon\}$ belongs to that level's <sigma-algebra>, so its <measure> is $\int\mathbf1_{\{Y_n\leq\varepsilon\}}Y_n\,d\lambda\leq\varepsilon$. 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 $\varepsilon$ for every $\varepsilon>0$, hence zero. If the same limit set has full $\lambda$-measure, the two <measures> are singular.
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