A Concise Introduction to the Theory of Integration
Designed for the full-time analyst, physicist, engineer, or economist, this book attempts to provide its readers with most of the measure theory they will ever need. The author has consistently developed the concrete rather than the abstract aspects of topics treated. The major new feature of this third edition is the inclusion of a new chapter in which the author introduces the Fourier transform. Solutions to all problems are provided. As a self-contained text, this book is excellent for both self-study and the classroom.
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The Classical Theory
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7r-system absolutely continuous addition algebra apply assertion assume Borel choose clear Clearly closed conclude continuously differentiable Corollary cr-algebra cubes Q define denote Dominated Convergence Theorem element exact cover example Exercise exists fact Fatou's Lemma fi(T Finally finite measure follows Fourier given Hence Hilbert space Holder's inequality holds intervals ip(x l,oo Lebesgue integral Lebesgue measure Lebesgue's Dominated Convergence Let G linear lp(m LP(RN Markov's inequality measurable function measure space metric space Monotone Convergence Theorem Moreover mutually disjoint non-decreasing functions non-empty non-negative measurable non-overlapping observe obvious open set open subset orthogonal orthonormal basis particular preceding prove R-valued rectangle respect result Riemann integrable right-continuous satisfies simple functions subspace suffices suppose Tonelli's Theorem translation invariance uniformly unique vanishes x-integrable x-measure