The book is devoted to the structural analysis of vector and random (or both) valued countably additive measures, and used for integral representations of random fields. The spaces can be Banach or Frechet types. Special attention is given to Bochner's boundedness principle and Grothendieck's representation unifying and simplyfying stochastic integrations. Several stationary aspects, extensions and random currents as well as related multilinear forms are analyzed, whilst numerous new procedures and results are included, and many research areas are opened up which also display the geometric aspects in multi dimensions.Contents:Introduction and MotivationSecond Order Random Measures and RepresentationsRandom Measures Admitting ControlsRandom Measures in Hilbert Space: Specialized AnalysisMore on Random Measures and IntegralsMartingale Type Measures and Their IntegralsMultiple Random Measures and IntegralsVector Measures and IntegralsRandom and Vector MultimeasuresReadership: Senior graduate students in probability and abstract analysis, mathematicians and statisticians.