Maybe I should be clearer: the Radon-Nikodym theorem is about the existence of a density between two absolutely continuous measures. In finance, the measures could be, for example, the risk-neutral measures corresponding to different numeraires (e.g. foreign and domestic bonds), or the physical measure and the risk-neutral measure.What is the density? it is basically a measure of how distorted probabilities of certain events become by moving from one measure to another. In finance, it is often given by a stochastic exponential. Girsanov's theorem tells you how dynamics of processes change when you change measure by a Radon-Nikodym density which is a stochastic exponential.In finance, you work on probability spaces which are complicated - if you work on the sample path space (Wiener space) then it is infinite-dimensional etc. etc. so obviously there are technicalities in proving the Radon-Nikodym theorem and Girsanov's theorem etc. On the other hand, if you want to intuitively understand what a Radon-Nikodym derivative/density is, it is easy. Think of the simplest (non-trivial) probability space and simplest measure you can. For me, this is tossing a coin. There are two outcomes: H or T. Now pick two probability measures on this space, which don't assign either outcome 0. In the example below, I picked a fair coin and a biased coin. Then to calculate the Radon-Nikodym derivative between the two measures, it is just the ratio of the respective probabilities. (This is also known as the likelihood ratio!)It is completely clear what the Radon-Nikodym derivative is and what, intuitively, it means.As with everything else in maths, once you understand it is easy. Thinking about R-N theorem for more complicated probability spaces and measures is more technical, but the idea is the same.