RadonNikodým

RadonNikodým is a concept in real analysis, specifically in measure theory. It is named after the mathematicians Jan Radon and Otto Nikodým. The Radon-Nikodým theorem provides a way to express a measure that is absolutely continuous with respect to another measure as an integral with respect to that other measure.

More formally, let $\mu$ and $\nu$ be two sigma-finite measures defined on a measurable space $(X, \mathcal{F})$. If $\nu$ is absolutely continuous with respect to $\mu$ (written as $\nu \ll \mu$), then there exists a non-negative, $\mathcal{F}$-measurable function $f: X \to [0, \infty)$ such that for every measurable set $A \in \mathcal{F}$, the measure of $A$ under $\nu$ is given by the integral of $f$ over $A$ with respect to $\mu$: $\nu(A) = \int_A f \, d\mu$. This function $f$ is called the Radon-Nikodým derivative or Radon-Nikodým density of $\nu$ with respect to $\mu$, and it is often denoted by $\frac{d\nu}{d\mu}$.

The theorem also states that this derivative $f$ is unique up to sets of $\mu$-measure zero. The existence of such a function $f$ is guaranteed under the condition that $\nu$ is absolutely continuous with respect to $\mu$ and that both measures are sigma-finite. This theorem is fundamental in many areas of mathematics, including probability theory, functional analysis, and stochastic calculus, as it allows for the transition between different probability distributions or measures.