RadonNikodymTheorem

The Radon-Nikodym derivative (or Radon-Nikodym theorem) is a central result in measure theory and probability theory that deals with the representation of one measure in terms of another. It's often used in the change of variables formula for integrals. Essentially, it says that under certain conditions, a measure can be expressed as an integral with respect to another measure.

Given two measures $\mu$ and $\nu$ on a measurable space $(X, \Sigma)$, it is said that $\mu$ is absolutely continuous with respect to $\nu$ (denoted $\mu << \nu$) if for every measurable set $A$ in $\Sigma$, $\nu(A) = 0$ implies $\mu(A) = 0$. This means that $\mu$ doesn't assign positive mass to $\nu$-negligible sets.

The Radon-Nikodym theorem states that if $\mu$ is absolutely continuous with respect to $\nu$, then there exists a measurable function $f: X \rightarrow [0, \infty)$ such that for all measurable sets $A$ in $\Sigma$, we have:

$$\mu(A) = \int_A f d\nu$$

Here, $\int_A f d\nu$ denotes the integral of $f$ with respect to $\nu$ over the set $A$. The function $f$ is called the Radon-Nikodym derivative of $\mu$ with respect to $\nu$, and it's usually denoted $d\mu/d\nu$.

To give a full proof of the Radon-Nikodym theorem requires some deep results from measure theory and real analysis, including the Hahn decomposition theorem and the Lebesgue monotone convergence theorem. But the essence of the proof is to build up the function $f$ by approximating the measure $\mu$ in a clever way, and then using properties of measure and integration to show that this approximation actually gives the desired result.

This is a very abstract mathematical concept that's used in many branches of mathematics, including probability theory (where measures often represent probabilities), functional analysis, and mathematical physics. The Radon-Nikodym derivative is especially important in the theory of stochastic processes, where it's often used to change from one probability measure to another (a process known as Girsanov's theorem) which is central to the risk-neutral measure in mathematical finance.