Lemma: Let $P$ and $\mu$ be probability measures on a measurable space $(\Omega, \mathcal{A})$ with restrictions $P_{m}$ and $\mu_{m}$ to the elements of an increasing sequence of σ-fields $\mathcal{A_{1}}⊂\mathcal{A_{2}}⊂···⊂\mathcal{A}$ that generates $\mathcal{A}$. If $P_{m}<<\mu_{m}$ for every $m$, then $dP_{m}/dμ_{m}→dP^{a}/dμ$, almost surely $[\mu]$, for $P^{a}$ the part of $P$ that is absolutely continuous with respect to $\mu$. Furthermore,this convergence is also in $L1(\mu)$ if and only if $P<<μ$
Additional information given from the proof of the Lemma. It states that if $P<<\mu$ then $dP_{m}/dμ_{m}$ is the conditional expectation $\mathbb{E}_{\mu}[dP/dμ|\mathcal{A}_{\mu}]$ and a uniformly integrable $\mu$-martingale that converges to $dP/dμ$.
Later based on this Lemma and on a partition of a space $X$, $(A_{1},...,A_{k})$, it states that a sequence of densities $p_{m}$ can be defined as $p_{m}=dP_{m}/d\mu_{m},$ where $P$ and $\mu$ follow the conditions of the Lemma, and also that the density of $P$ is given as
$$p_{m}=\sum \frac{P(A_{i})}{\mu(A_{i})}1_{A_{i}}$$
Those are notes from the book "Fundamentals of Nonparametric Bayesian Inference".
The Lemma can be found on "L - Miscellaneous Results" while the statment of the density expression for the partition corresponds to the Theorem 3.16 of Chapter 3.
Without any further assumption this is surely false. Consider the RND of standard Gaussian measure w.r.t. Lebesgue measure and the partition $(-\infty, 0], (0,\infty)$. If what you are saying is true then the Guassian density function must be a constant on each of the intervals $(-\infty, 0], (0,\infty)$!.