In this chapter, we introduce the second of the two geometric objects that we associate with cake division. We call this object the Radon—Nikodym Set, or RNS. For our first geometric object, the IPS (or, more generally, the FIPS), we were interested in a geometric perspective on the set of all partitions of C. Our present goal is quite different. When we introduced the IPS, we started with the cake C, we considered the set of all partitions of C, and then we formed a geometric object, the IPS (or the FIPS), that contains useful information about this set. Now, we start with the cake C, we form a geometric object, the RNS, and then we use this new geometric object to construct partitions having desired properties. In the next chapter, we will use the RNS to obtain a new characterization of Pareto maximality and Pareto minimality. In Chapter 12, we will study the relationship between the IPS and the RNS.

In Section 9A, we assume that the measures are absolutely continuous with respect to each other. In Section 9B, we consider what happens when absolute continuity fails. Much of the material in this chapter is attributable to D. Weller ([43]).