Measures of information for continuous observations

Certain measures of information between a state process and an observation process and between a state process and an estimate of the state process are considered. The observations are assumed to have sample paths which are almost surely continuous. The calculus of martingales and probability measure transformations are used to determine representations of the observation process and the state estimate on different probability spaces obtained by a change in the sub-o-algebra and/or the probability measure. The Radon-Nikodym derivatives required to calculate the information theoretic quantities are obtained from these representations.