Let $(\Omega,\mathscr{A},\mathbf{P} )$ be a probability space and $$\xi :(\Omega,\mathscr{A},\mathbf{P} )\longmapsto(\mathbb{R},\mathcal{B}(\mathbb{R}))$$ is a random variable, with $$\mathbf{Q}(B)=\mathbf{P}(\xi^{-1}(B))\;\;\forall B \in \mathcal{B}(\mathbb{R}) $$ distribution, where $\mathbf{Q}\ll\mathbf{P}$.
I was wondering, in this case what is the $\frac{d\mathbf{Q}}{d\mathbf{P}}$ Radon-Nykodim derivative?
To understand it clearly, I also want to ask about the following statement whether it is true or not: If $\lambda$ is the Lebesgue-measure, and $\mathbf{Q}\ll\lambda$, then $\frac{d\mathbf{Q}}{d\lambda}=f$ is the ordinary probability density function?
I find it true by definition, because $$\mathbf{Q}\left(B\right)=\int_{B}fd\lambda.$$ Am I right?