Questions tagged [radon-nikodym]

Let $\mu_1$ and $\mu_2$ and $\nu$ be two finite measures on the measurable space $(\Omega, \mathcal{S})$ such that $\nu \ll \mu_i$, $i = 1, 2$, and $f_i = \frac{d\nu}{d\mu_i}$. Show that then $\nu \ll \mu_1 + \mu_2$ and $$ \frac{d\nu}{d(\mu_1 +…

The Wikipedia proof of Radon-Nikodym first shows for positive measures, then extends to real measures by applying the positive measure Radon-Nikodym theorem to the unique + and - (positive) measures obtained via Hahn-Jordan decomposition. But one…

Let $(X_t)$ be an Ito diffusion with speed $(V_t)$, under a probability measure P. Could there exist a change of measure to a probability measure Q, with Q ~ P, under which $(X_t)$ is an Ito diffusion with a different speed $(V'_t)$? I've figured…

I am reading An Informal Introduction to Stochastic Calculus with Applications and I've come across this statement of the Radon-Nikodym theorem. "Consider the probability space $(\Omega ,\mathcal {F},P)$, and let $\mathcal {G}$ be a $\sigma$ field…