1 Radon-Nikodym Derivative
The Radon-Nikodym derivative is a measurable function relating two probability measures defined on the same measurable space. Specifically, the Radon–Nikodym derivative tells us how to reweight probabilities from measure \(\mathbb{P}\) to measure \(\mathbb{Q}\).
For probability measures \(\mathbb{P}\) and \(\mathbb{Q}\) defined on measurable space \((\Omega, \mathcal{F})\) assume absolute continuity \(\mathbb{Q}\ll \mathbb{P}\). The Radon-Nikodym Theorem guarantees the existence of a measurable function \(Z:\Omega \to[0,\infty)\) known as the Radon-Nikodym (RN) derivative denoted \[ Z= \frac{d\mathbb{Q}}{d\mathbb{P}} \] such that \[ \mathbb{Q}(A)=\int_{A}^{}{Z}~d{\mathbb{P}},\quad \forall A \in \mathcal{F}. \]
Recall that two measures \(\mathbb{P}\) and \(\mathbb{Q}\) are said to be absolutely continuous, denoted \(\mathbb{Q}\ll \mathbb{P}\) if \(\mathbb{P}(A)=0 \implies \mathbb{Q}(A)=0\) for all \(A\in \mathcal{F}\).
To gain an intuitive understanding of the role of the RN derivative note how we can rewrite the expectation of a random variable \(X\) under \(\mathbb{Q}\) and then \(\mathbb{P}\) \[ \mathbb{E}^{\mathbb{Q}}[X]= \int_{\Omega}^{}{X}~d{\mathbb{Q}}=\int_{\Omega}^{}{X\cdot Z}~d{\mathbb{P}}=\mathbb{E}^{\mathbb{P}}[X]. \] The main application of the RN derivative is in mathematical finance where it is used to derive the risk-free measure used to price derivatives.