Lectures on Real Analysis: textbook

The book is devoted to the basics of integration theory and related topics. Particular attention is paid to the theory of the integral as Lebesgue. The replacement of variables in a multiple integral and the construction of the measure on the surface are discussed. Annexes of general theory, harmonic functions, asymptotic formulas of Laplace, rows and Fourier transform, formulas associated with the stationary phase method, etc. All themes are set out for functions of one and several variables. Actual sections of real analysis (for example, Hausdorff"s measures) are considered, important geometric applications: the inequality of Brunna-Minkowski, the brower theorem, the theorem on continuous vector fields on the sphere, etc. The questions of functional analysis are addressed. The presentation is based on lectures that have repeatedly read by the authors at the Mathematical and Mechanical Faculty of St. Petersburg State University. The book contains more than 600 examples and exercises