Reseña

Although mankind is measuring things from very ancient times, the object of this book dates back to the thesis of Lebesgue, 1902. The book solves to problems: PROBLEM 1. Given a positive integerp, is there a not identically null, non-negative function defined on the bounded sets of RP which is invariant under translation and such that its value on a finite union of pairwise disjoint bounded sets is equal to the addition of the values of all of them? PROBLEM 2. For p = 1 and 2, is there a function as in Problem 1 but replacing the invariance under translation by the more general invariance under isometries? Through the book the tools necessary to answer these two problems are developed. The book is organized in five chapters ended by a solved exercise related with its contents that serves to fix concepts. The book is well written, concepts are clearly introduced, and the proofs of theorems can be understood as easily as possible.