The main goal of this Encyclopaedia is to survey Measure Theory with its many different branches and its relations with other areas of Mathematics. The aim of the Encyclopaedia is to cover New Fields, Approaches and Applications which support the idea of "Measure" in a wider sense. Although chapters are written of surveys in the various areas they contain many special topics and Challenging Problems valuable for Experts and Rich sources of Inspiration. Mathematicians from other areas as well as Physicists, Computer Scientists, Engineers and Econometrists will find useful results and powerful methods for their research. The reader may find in the Encyclopaedia many close relations to other Mathematical Areas: Real Analysis, Probability Theory, Statistics, Ergodic Theory, Functional Analysis, Potential Theory, Topology, Set Theory, Geometry, Differential Equations, Optimization, Variation Analysis, Decision making and others. The Encyclopaedia is a rich source of Relevant References to Articles, Books and Lecture Notes and it contains for the reader's convenience an extensive subject.