This book, "Gaussian Measures in Banach Spaces," is a monograph based on lecture notes from a course on "Applications of Measure Theory" given at the University of Virginia in 1974. The course aimed to introduce the concept of abstract Wiener space and explore related topics. The content covers the first two chapters and the first three sections of Chapter III, with the last four sections added later. The author regrets not discussing recent works by J. Eells, K.D. Elworthy, and R. Ramer on the integration of Banach manifolds. The author thanks Leonard Gross, Kiyosi Ito, and Tavan Trent for their contributions. The manuscript was typed by Barbara Smith and Fukuko Kuo, and the preparation was partially supported by the National Science Foundation. The book is divided into three chapters. Chapter I discusses Gaussian measures in Banach spaces, including Hilbert-Schmidt and trace class operators, Borel measures in Hilbert space, Wiener measure, abstract Wiener space, and related theorems. Chapter II covers equivalence and orthogonality of Gaussian measures, including translation of Wiener measure, Kakutani's theorem, Feldman-Hajek's theorem, and transformation formulas. Chapter III presents results about abstract Wiener space, including Gaussian measures in Banach spaces, a probabilistic proof of a theorem, integrability of certain functions, and potential theory. The book also includes a stochastic integral, divergence theorem, and comments on each chapter. References and an index are provided. The book is intended for mathematicians and researchers in the field of measure theory and stochastic processes.This book, "Gaussian Measures in Banach Spaces," is a monograph based on lecture notes from a course on "Applications of Measure Theory" given at the University of Virginia in 1974. The course aimed to introduce the concept of abstract Wiener space and explore related topics. The content covers the first two chapters and the first three sections of Chapter III, with the last four sections added later. The author regrets not discussing recent works by J. Eells, K.D. Elworthy, and R. Ramer on the integration of Banach manifolds. The author thanks Leonard Gross, Kiyosi Ito, and Tavan Trent for their contributions. The manuscript was typed by Barbara Smith and Fukuko Kuo, and the preparation was partially supported by the National Science Foundation. The book is divided into three chapters. Chapter I discusses Gaussian measures in Banach spaces, including Hilbert-Schmidt and trace class operators, Borel measures in Hilbert space, Wiener measure, abstract Wiener space, and related theorems. Chapter II covers equivalence and orthogonality of Gaussian measures, including translation of Wiener measure, Kakutani's theorem, Feldman-Hajek's theorem, and transformation formulas. Chapter III presents results about abstract Wiener space, including Gaussian measures in Banach spaces, a probabilistic proof of a theorem, integrability of certain functions, and potential theory. The book also includes a stochastic integral, divergence theorem, and comments on each chapter. References and an index are provided. The book is intended for mathematicians and researchers in the field of measure theory and stochastic processes.