I am familiar only with the most basic notions and intuitions of measure theory, and am seeking some clarification about the following definition of a measure:
Let $\nu$ be a measure on $([0,1], \mathcal{B}([0,1])$ with $\nu(d\pi) = \alpha\pi^{-1}(1-\pi)^{\alpha-1}d\pi$.
I'm trying to understand the definition of the given $\nu(d\pi)$ here. The best interpretation I can think of comes from the definition of the integral with respect to a measure:
$$ \nu(A) = \int_{\pi\in[0,1]}\mathbb{I}_{A}(\pi)\nu(d\pi)=\int_{A}\alpha\pi^{-1}(1-\pi)^{\alpha-1}d\pi \quad\quad\quad A \in \mathcal{B}([0,1]). $$
Letting $\lambda$ denote the Lebesgue measure onf $\mathbb{R}$, we have $\lambda(d\pi)=d\pi$ which means the notation suggests $\nu << \lambda$ so that $\nu(A) = \int_{A}gd\lambda$ with density $g(\pi) = \alpha\pi^{-1}(1-\pi)^{\alpha-1}$. Is this interpretation correct?
It seems we can generalize this notation so that, for some other measure $\mu$, we can define $\nu(d\pi) = \alpha\pi^{-1}(1-\pi)^{\alpha-1}\mu(d\pi)$, which gives the density of $\nu<<\mu$.