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Two measures $\mu$ and $\nu$ are singular to each other if there exists a measurable set $A$ such that $\mu(A) = 0$ and $\nu(A^c) = 0$. Singular measures do not have a Radon-Nikodym derivative with respect to each other.

If $\nu$ is absolutely continuous with respect to $\mu$, then there exists a $\mu$-integrable function $f$, such that for any measurable set $A$, $\nu(A) = \int_A f d\mu$. The function $f$ is called the Radon-Nikodym derivative of $\nu$ with respect to $\mu$, often denoted as $\frac{d\nu}{d\mu}$.

In probability theory, the Radon-Nikodym theorem allows us to find the density of one probability measure with respect to another. This is particularly useful for changing probability measures or determining how likelihoods change under new measures.

The Radon-Nikodym derivative is unique up to sets of $\mu$-measure zero. If $f$ and $g$ are both Radon-Nikodym derivatives of $\nu$ with respect to $\mu$, then $f = g$ $\mu$-almost everywhere.

Given two $σ$-finite measures $\mu$ and $\nu$, there exists a unique pair of measures $\nu_{ac}$ and $\nu_s$ such that $\nu = \nu_{ac} + \nu_s$, where $\nu_{ac}$ is absolutely continuous with respect to $\mu$ and $\nu_s$ is singular with respect to $\mu$. This theorem is essential for the application of the Radon-Nikodym theorem.

If $X$ is an integrable random variable with respect to $\mu$, and $\frac{d\nu}{d\mu}$ exists, the expected value of $X$ with respect to $\nu$ can be calculated as $\mathbb{E}_{\nu}[X] = \int X \frac{d\nu}{d\mu} d\mu$.