Till now I have only seen Radon Nikodyn derivative in context of measures.But I am reading a book which says "For continuous processes A and B which are of bounded variation absolute continuity implies that the Radon-Nikodyn derivative can be chosen to be a predictable process ".Can anyone explain the Radon-Nikodyn derivative in this setting.The lemma is stated as follows in the book Lemma
Also what is the meaning of the the notation $$dA(\omega)<<dB(\omega)$$
If $A_t$ is a continuous processes of bounded variation, it induces a measure $\mu_A$ on $\mathbb{R}_+$ defined by $$\mu_A(E) := \int_E dA_s.$$ Then the Radon-Nikodym derivative of $B$ with respect to $A$ refers to the Radon-Nikodym derivative of the respective induced measure.
The notation $dA(\omega) << dB(\omega)$ is typically written $dA(\omega) \ll dB(\omega)$ and means that the measure induced by $A$ is absolutely continuous with respect to the measure induced by $B$, i.e. if $\int_E dB(\omega) = 0$ then $\int_E dA(\omega) = 0$.