Let $\mu$ be a positive Radon probability measure, $\nu$ be a $\sigma-finite$ measure on a measurable space $X$. Then $\mu$ can be decomposited into absolutely part $\mu^a$ and singular part $\mu^s$ w.r.t $\nu$.
Let $C\subset X$ be the subset where $\mu^s$ is concentrated. Then is $C$ $\mu$-measurable? Is $C$ a Borel subset?
First of all, you will need to clarify what do you mean by concentration. Usually we say that $\mu$ is supported on $A$ if $\mu(X\setminus A) = 0$. Clearly, $\mu^s$ is supported on $X$ which is Borel by definition. Now, if you would like to narrow the definition of concentration, the only good candidate I know is support of measure, which is more or less the minimal set on which the measure is concetrated. However, the support of measure is always a closed set, hence Borel.