I TAKEUTIlZARING. Introduction to Axiomatic Set Theory. 2nd ed. OXTOBY. Measure and Category. 2nd ed. SCHAEFFER. Topological Vector Spaces. HILTON/STAMMBACH. A Course in Homological Algebra. MAC LANE. Categories for the Working Mathematician. HUGHES/PIPER. Projective Planes. SERRE. A Course in Arithmetic. T AKEUTIlZARING. Axiomatic Set Theory. HUMPHREYS. Introduction to Lie Algebras and Representation Theory. to COHEN. A Course in Simple Homotopy Theory. II CONWAY. Functions of One Complex Variable. 2nd ed. BEALS. Advanced Mathematical Analysis. ANDERSON/fuLLER. Rings and Categories of Modules. GOLUBITSKy/GUILLEMIN. Stable Mappings and Their Singularities. BERBERIAN. Lectures in Functional Analysis and Operator Theory. WINTER. The Structure of Fields. ROSENBLATT. Random Processes. 2nd ed. HALMOS. Measure Theory. HALMOS. A Hilbert Space Problem Book. 2nd ed., revised. HUSEMOLLER. Fibre Bundles. 2nd ed. HUMPHREYs. Linear Algebraic Groups. BARNES/MACK. An Algebraic Introduction to Mathematical Logic. GREUB. Linear Algebra. 4th ed. HOLMES. Geometric Functional Analysis and its Applications. HEwm/STRoMBERG. Real and Abstract Analysis. MANES. Algebraic Theories.

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Abstract. We analyze a class of weakly differentiable vector fields F : R N → R N with the property that F ∈ L ∞ and div F is a Radon measure. These fields are called bounded divergencemeasure fields. The primary focus of our investigation is to introduce a suitable notion of the normal trace of any divergence-measure field F over the boundary of an arbitrary set of finite perimeter, which ensures the validity of the Gauss-Green theorem. To achieve this, we first develop an alternative way to establish the Gauss-Green theorem for any smooth bounded set with F ∈ L ∞ . Then we establish a fundamental approximation theorem which states that, given a Radon measure µ that is absolutely continuous with respect to H N−1 on R N , any set of finite perimeter can be approximated by a family of sets with smooth boundary essentially from the measure-theoretic interior of the set with respect to the measure µ . We employ this approximation theorem to derive the normal trace of F on the boundary of any set of finite perimeter, E, as the limit of the normal traces of F on the boundaries of the approximate sets with smooth boundary, so that the Gauss-Green theorem for F holds on E.With these results, we analyze the Cauchy fluxes that are bounded by a Radon measure over any oriented surface (i.e. an (N − 1)-dimensional surface that is a part of the boundary of a set of finite perimeter) and thereby develop a general mathematical formulation of the physical principle of balance law through the Cauchy flux. Finally, we apply this framework to the derivation of systems of balance laws with measurevalued source terms from the formulation of balance law. This framework also allows the recovery of Cauchy entropy fluxes through the Lax entropy inequality for entropy solutions of hyperbolic conservation laws.

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