The Radon-Nikodym Theorem for a Finite Measure and a Sigma-Finite Measure, and Uniqueness of Densities

In the setting of Measure Spaces and the Lebesgue Integral: Standing Notation, let $(X,\mathcal{F},\mu)$ be a measure space and let $\nu$ be a finite measure on $(X,\mathcal{F})$. For a measurable $h:X\to\mathbb{R}$ with $0\le h(x)$ for every $x\in X$, the measure with density $h$ with respect to $\mu$ is that of claim 3 of that lemma, and $h$ is called a density of $\nu$ with respect to $\mu$ if that measure is $\nu$, that is, if

1. (Uniqueness)¶ Let $h_{1}$ and $h_{2}$ be densities of $\nu$ with respect to $\mu$. Then $h_{1}$ and $h_{2}$ are integrable with $\int_{X}h_{1}\,d\mu=\int_{X}h_{2}\,d\mu=\nu(X)$, and the set $\{x\in X:h_{1}(x)\ne h_{2}(x)\}$ belongs to $\mathcal{F}$ and has $\mu$-measure $0$.

2. (Existence)¶ Suppose that $\mu$ is $\sigma$-finite and that $\nu(A)=0$ for every $A\in\mathcal{F}$ with $\mu(A)=0$. Then $\nu$ has a density with respect to $\mu$.