In this chapter we introduce some elementary concepts from measure theory as a start in our examination of the dynamical origin of increasing entropy. Section A draws a connection between thermodynamic systems and measure spaces. Section B briefly considers dynamics and phase spaces, while Section C introduces the notion of a density and postulates that the state of a thermodynamic system is characterized by a density. In Section D we introduce the Boltzmann-Gibbs entropy, and prove that it is the unique (up to a multiplicative constant) entropy definition that satisfies the physical requirement (additivity) that entropy be an extensive quantity.
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