Titlebook: Real Analysis: Measures, Integrals and Applications; Boris Makarov,Anatolii Podkorytov Textbook 2013 Springer-Verlag London Ltd., part of

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Lament

Measurable Functions,t types of convergence for sequences of measurable functions. We prove the important theorem of Egorov, which reduces pointwise convergence to uniform convergence by deleting an appropriate set of arbitrarily small measure. We discuss the question of approximation of measurable functions by continuo

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The Integral,he countable additivity and absolute continuity of the integral. In Sects. . and ., we discuss the properties of the integral for functions of one and several variables separately. In the next section, we thoroughly study conditions under which the passage to the limit under the integral sign is pos

青石板

The Product Measure,enting the Lebesgue measure in a multi-dimensional space as a product of measures and prove Cavalieri’s principle. We give several examples of application of the results obtained. In particular, we obtain a formula connecting the Euler and Γ functions and prove the Gagliardo–Nirenberg–Sobolev inequa

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Change of Variables in an Integral,sis. Sect. . introduces the notion of weighted image of a measure. A number of important particular cases is then explored based on this notion. They are: change of variables in the Lebesgue integral under a diffeomorphism, the calculation of integrals using the distributions on the line correspondi

Musket

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Surface Integrals,cuss its basic properties (including the one-dimensional case of the length of a curve) and different approaches to the definition of the area of a surface (Schwarz lantern)..In Sects. . and ., we establish formulas for computing the area of a surface and integrals over it, including the Kronrod–Fed

Adenoma

Approximation and Convolution in the Spaces ,,imation “in the mean”..Preliminarily, in Sect. . we introduce a family of .-metrics and establish basic properties of the spaces . for 1⩽.⩽∞. In Sect. ., we study approximations with respect to the .-metric by continuous and smooth functions. We also prove that summable functions are continuous in m

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