ABSTRACT
In this chapter, the author examines the structure of a charge λ on a σ- algebra S. The proof of the theorem relies heavily on the arguments used in The Jordan Decomposition of a Finite Charge Theorem and so it is easy to just nod readers head wisely and take our word for it. But it is always important to do the work for ourself. The author shows that any charge A has associated positive and negative sets. The author looks at absolute continuity in the context of charges. The author knows that integrals of summable functions define charges which are absolutely continuous with respect to the measure we are using for the integration. The converse of this is that if a measure is absolutely continuous, the author finds a summable function so that the measure can be found by integration. The author proves the Radon- Nikodym theorem for the case that λ is a sigma-finite charge.
