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Measure Theory and Integration

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Measure Theory and Integration
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  1. Beskrivelse

    Offers a treatment of measure and integration that begins with a brief review of the Riemann integral and proceeds to a construction of Lebesgue measure on the real line. This text also treats probabilistic concepts, in chapters on ergodic theory, probability spaces and random variables, Wiener measure and Brownian motion, and martingales. Illustrations

  2. Yderligere info
    Udgivelsesdato:
    15-08-2006
    Sprog:
    Engelsk
    ISBN13:
    9780821841808
    Sidetal:
    319
    Vægt:
    765 g
    Højde:
    22 mm
    Nummer i serien:
    No. 76
    Mærkat:
    Bog, hardback
    Format:
    Hardback
    • Bibliotekernes beskrivelse

      This self-contained treatment of measure and integration begins with a brief review of the Riemann integral and proceeds to a construction of Lebesgue measure on the real line. From there the reader is led to the general notion of measure, to the construction of the Lebesgue integral on a measure space, and to the major limit theorems, such as the Monotone and Dominated Convergence Theorems. The treatment proceeds to $L^p$ spaces, normed linear spaces that are shown to be complete (i.e., Banach spaces) due to the limit theorems. Particular attention is paid to $L^2$ spaces as Hilbert spaces, with a useful geometrical structure. Having gotten quickly to the heart of the matter, the text proceeds to broaden its scope.There are further constructions of measures, including Lebesgue measure on $n$-dimensional Euclidean space. There are also discussions of surface measure, and more generally of Riemannian manifolds and the measures they inherit, and an appendix on the integration of differential forms. Further geometric aspects are explored in a chapter on Hausdorff measure. The text also treats probabilistic concepts, in chapters on ergodic theory, probability spaces and random variables, Wiener measure and Brownian motion, and martingales. This text will prepare graduate students for more advanced studies in functional analysis, harmonic analysis, stochastic analysis, and geometric measure theory.

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