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Measure theory
I/.SS,R
Cornput. Maths. Math. Phys.
Vol. 22,
No. 4,
pp. 258-259,
Printedin Great Britain
1982.
0041-5553/82$07.50+.00
01983. PergamonPressLtd.
BOOK REVIEW *
D. COHN, Measure theory, Birkhauser, Boston etc., 1980.
THE AUTHOR tells us in the introduction that his aim is to treat measure and integral theory to the extent needed for applications to analysis and probability theory. The book has nine chapters and five appendices. Chapter I: Measures. The concepts of algebra and u-algebra Sz of subsets of a given set X are introduced, and on their basis the finite and u-finite measure in d, and a space with measure (X, &, p) are defined. Bore1 and Lebesgue measures are considered as examples. On the basis of the axioms of choice, the existence of Lebesgue-unmeasurable sets in the interval [0,1] is proved. The concepts of perfect and regular measure are introduced. Chapter II: Functions and integrals. With the aid of measurable space (X, &) the concept of p-measurable function f:A + [--,t -1 is introduced, where A is any set of .QZ.The concept of an integral is extended to non-negative p-measurable functions. The basic properties of the integral are proved, including the theorems of Levi, Lebesgue, and Fatou on passage to the limit under the integral sign. The Riemann and Lebesgue integrals are compared. Chapter III: Convergence. The concept of convergent with respect to measure p sequence of p-measurable functions is introduced, and Egorov’s theorem is proved, Spaces L P (X, & p), are defined, and the Holder and Minkowski inequalities are proved. Chapter IV: Alternating and complex-valued measures. The Hahn and Jordan theorems on representation of alternating measures in terms of positive measures are proved. The concept of absolute continuity of measure p with respect to measure v is introduced, and the Radon-Nikodym theorem on representation of measure p in terms of measure u is proved. The concept of pabsolute continuity of a function is then introduced, and Lebesgue’s theorem is proved, to the effect that the class of absolutely continuous functions is the same as the class of indefinite Lebesgue integrals of integrable functions. Chapter V: Measures on Cartesian products of sets. The main result is Fubini’s theorem on reduction of a multiple to a repeated integral. Chapter VI: Differentiation. Section 1 deals with linear replacement of the variables under the n-tuple integral sign, and Section 2 with differentiation of measures, the main result being Vitali’s theorem on coverings of sets in Rn,The differentiability of functions is discussed in Section 3 and Lebesgue’s theorems on the differentiability of functions of class L1 (R), and on restoration of a differentiable function from its derivative are proved.
*Z/L vjkhisl Mat. mat. Fiz., 22,4,
1016-1017,
1982.
258
259
Book reviews
Chapter VII: Measures on locally compact spaces. The concepts of compact and locally compact topological space are introduced. There is a proof of Riesz’s theorem on the representation of a positive linear functional in set C Q of continuous functions f in a locally compact topological Hausdorff space X as the integral of the functionfwith respect to a Bore1 measure in X; and Luzin’s theorem on the C-property of kmeasurable functions in X is proved. Cnapter VIII: Polish spaces and analytic sets. The author defines a Polish space (P.S.) as a topological space which can be metrized by means of a complete metric. The elementary properties of P.S. are considered in Section 1. The concept of analytic set is introduced. Examples and the elementary properties of analytic sets are given. The theorem on separability of analytic sets and some corollaries are stated. In Section 4 it is proved that every analytic set of a P.S. X is p-measurable, if p is a finite Bore1 measure in X. The properties of standard, analytic, Luzin, and Suslin spaces are examined. It is shown that every Bore1 measure in a Suslin space is regular. A measurable space (X, ~4) is called standard if there exists a P.S.Z such that (X, &) is isomorphic to (2, a(z)), where 3J (2) is the Bore1 u-algebra of space Z. If there exists an analytic subset A of P.S.Z such is isomorphic to (X,at), then space (X,at) is called analytic. that (A, J(A)) Chapter IX: Haar measure. Left and right Haar measures in a locally compact group are defined. It is shown that, in every locally compact group G, there exists a left Haar measure, and all the left Haar measures in G differ only by a positive factor. The properties of a Haar measure in a locally compact group G are studied, as are the properties of the algebras Ll (G) and M(G) of, respectively, functions integrable in G with respect to a left Haar measure, and essentially bounded measurable functions in G. As the multiplication operation, the convolution of two functions, respectively of L 1 (G) and M(G) is considered. There are numerous exercises at the end of each section. Appendix A: Notation and theory of sets. B: Algebra. C: Analysis and topology in Rn. D: Topological and metric spaces. E: The Bochner integral. Some facts concerning these topics are given in the Appendices, mainly without proof. B. Golubov Translated by D. E. Brown
Cornput. Maths. Math. Phys.
Vol. 22,
No. 4,
pp. 258-259,
Printedin Great Britain
1982.
0041-5553/82$07.50+.00
01983. PergamonPressLtd.
BOOK REVIEW *
D. COHN, Measure theory, Birkhauser, Boston etc., 1980.
THE AUTHOR tells us in the introduction that his aim is to treat measure and integral theory to the extent needed for applications to analysis and probability theory. The book has nine chapters and five appendices. Chapter I: Measures. The concepts of algebra and u-algebra Sz of subsets of a given set X are introduced, and on their basis the finite and u-finite measure in d, and a space with measure (X, &, p) are defined. Bore1 and Lebesgue measures are considered as examples. On the basis of the axioms of choice, the existence of Lebesgue-unmeasurable sets in the interval [0,1] is proved. The concepts of perfect and regular measure are introduced. Chapter II: Functions and integrals. With the aid of measurable space (X, &) the concept of p-measurable function f:A + [--,t -1 is introduced, where A is any set of .QZ.The concept of an integral is extended to non-negative p-measurable functions. The basic properties of the integral are proved, including the theorems of Levi, Lebesgue, and Fatou on passage to the limit under the integral sign. The Riemann and Lebesgue integrals are compared. Chapter III: Convergence. The concept of convergent with respect to measure p sequence of p-measurable functions is introduced, and Egorov’s theorem is proved, Spaces L P (X, & p), are defined, and the Holder and Minkowski inequalities are proved. Chapter IV: Alternating and complex-valued measures. The Hahn and Jordan theorems on representation of alternating measures in terms of positive measures are proved. The concept of absolute continuity of measure p with respect to measure v is introduced, and the Radon-Nikodym theorem on representation of measure p in terms of measure u is proved. The concept of pabsolute continuity of a function is then introduced, and Lebesgue’s theorem is proved, to the effect that the class of absolutely continuous functions is the same as the class of indefinite Lebesgue integrals of integrable functions. Chapter V: Measures on Cartesian products of sets. The main result is Fubini’s theorem on reduction of a multiple to a repeated integral. Chapter VI: Differentiation. Section 1 deals with linear replacement of the variables under the n-tuple integral sign, and Section 2 with differentiation of measures, the main result being Vitali’s theorem on coverings of sets in Rn,The differentiability of functions is discussed in Section 3 and Lebesgue’s theorems on the differentiability of functions of class L1 (R), and on restoration of a differentiable function from its derivative are proved.
*Z/L vjkhisl Mat. mat. Fiz., 22,4,
1016-1017,
1982.
258
259
Book reviews
Chapter VII: Measures on locally compact spaces. The concepts of compact and locally compact topological space are introduced. There is a proof of Riesz’s theorem on the representation of a positive linear functional in set C Q of continuous functions f in a locally compact topological Hausdorff space X as the integral of the functionfwith respect to a Bore1 measure in X; and Luzin’s theorem on the C-property of kmeasurable functions in X is proved. Cnapter VIII: Polish spaces and analytic sets. The author defines a Polish space (P.S.) as a topological space which can be metrized by means of a complete metric. The elementary properties of P.S. are considered in Section 1. The concept of analytic set is introduced. Examples and the elementary properties of analytic sets are given. The theorem on separability of analytic sets and some corollaries are stated. In Section 4 it is proved that every analytic set of a P.S. X is p-measurable, if p is a finite Bore1 measure in X. The properties of standard, analytic, Luzin, and Suslin spaces are examined. It is shown that every Bore1 measure in a Suslin space is regular. A measurable space (X, ~4) is called standard if there exists a P.S.Z such that (X, &) is isomorphic to (2, a(z)), where 3J (2) is the Bore1 u-algebra of space Z. If there exists an analytic subset A of P.S.Z such is isomorphic to (X,at), then space (X,at) is called analytic. that (A, J(A)) Chapter IX: Haar measure. Left and right Haar measures in a locally compact group are defined. It is shown that, in every locally compact group G, there exists a left Haar measure, and all the left Haar measures in G differ only by a positive factor. The properties of a Haar measure in a locally compact group G are studied, as are the properties of the algebras Ll (G) and M(G) of, respectively, functions integrable in G with respect to a left Haar measure, and essentially bounded measurable functions in G. As the multiplication operation, the convolution of two functions, respectively of L 1 (G) and M(G) is considered. There are numerous exercises at the end of each section. Appendix A: Notation and theory of sets. B: Algebra. C: Analysis and topology in Rn. D: Topological and metric spaces. E: The Bochner integral. Some facts concerning these topics are given in the Appendices, mainly without proof. B. Golubov Translated by D. E. Brown