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 This book offers to the reader a self-contained treatment and systematic exposition of the real-valued theory of a nonabsolute integral on measure spaces. It is an introductory textbook to Henstock–Kurzweil type integrals defined on abstract spaces. It contains both classical and original results that are accessible to a large class of readers.It is widely acknowledged that the biggest difficulty in defining a Henstock–Kurzweil integral beyond Euclidean spaces is the definition of a set of measurable sets which will play the role of "intervals" in the abstract setting. In this book the author shows a creative and innovative way of defining "intervals" in measure spaces, and prove many interesting and important results including the well-known Radon–Nikodým theorem.