Theory of Functions of a Real Variable

Real-variable functional theory is an important component of analysis, a subject that learns between analytic mathematics and functional analytics.We study measure theory, integral theory, bounded variational function theory in this book.First, we study the properties of measure spaces, measurable sets, and measurable functions.Then we study the integral of measurable functions in measure space, its properties, the relationship between the Riemann integral and the Lebesgue integral, and so on.Finally we study the addition of generalized measure concepts to bounded variational functions such as additive set functions, absolute continuous functions and singular functions, and the Radon-Nicotheim theorem, the fundamental theorem of Lebesgue integrals, etc.This book is not only intended to be a textbook for students in mathematics, but also to be used as a reference for students, scientists and technicians who want to study mathematics more deeply and apply it to their specialties.We should study this subject well and prepare ourselves as able talents capable of successfully solving the scientific and technological problems arising in various sectors of the national economy.