Suppose we have some finite measure $\mu$ on the Borel sets of $R$. It is then assumed that there exists a density for this measure. What does that mean?
I am only familiar with densities in the measure theoretic probability setting: a random variable $X$ with distribution $P^X$ has density $f$ if $f$ is a positive measurable function and it holds for all Borel sets $B$ that $P^X(B)=\lambda(I_{B}f)$, where $\lambda$ is the Lebesgue measure.
In direct analogue, would it mean that there exists a positive measurable function $f$ such that $\mu(B)=\lambda(I_B f)$?