Basics.- Caratheodory's Construction and Lower-Dimensional Measures.- Invariant Measures and the Construction of Haar Measure..- Covering Theorems and the Differentiation of Integrals.- Analytical Tools: The Area Formula, the Coarea Formula, and Poincare Inequalities..- The Calculus of Differential Forms and Stokes's Theorem.- to Currents.- Currents and the Calculus of Variations.- Regularity of Mass-Minimizing Currents.
From the reviews:"This is a graduate textbook with the main purpose of introducing geometric measure theory through the notion of currents. ... One of the most important features of this text is that it is self-contained ... . The book also contains an Appendix ... as well as extended list of references, making it a good text for a graduate course, as well as for an independent or self study." (Mihaela Poplicher, The Mathematical Association of America, March, 2009)"The book under review succeeds in giving a complete and readable introduction to geometric measure theory. It can be used by students willing to learn this beautiful theory or by teachers as a basis for a one- or two-semester course." (Andreas Bernig, Mathematical Reviews, Issue 2009 m)"The authors present main fields of applications, namely the isoperimetric problem and the regularity of minimal currents. The exposition is detailed and very well organized and therefore the book should be quite accessible for graduate students." (R. Steinbauer, Monatshefte fur Mathematik, Vol. 162 (3), March, 2011)