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Book 3A

Measure, Integration and Lᵖ

An integral that behaves under limits

Lebesgue measure and integration, the convergence theorems, product measures and change of variables, Lᵖ spaces, and convolution, ending with the Gaussian integrals that run all the way to Perelman.

  • Main companionAn Introduction to Measure Theory · Terence Tao
  • OptionalReal Analysis · Stein & Shakarchi
In preparation

Contents

  1. 1
    The Problem of Measure

    What a size should be, why Jordan measure is not enough, and sets that cannot be measured.

    In preparation

  2. 2
    Lebesgue Measure

    Outer measure, measurable sets, null sets and the Cantor set.

    In preparation

  3. 3
    The Lebesgue Integral

    Monotone and dominated convergence, Fatou’s lemma, and how mass escapes in a limit.

    In preparation

  4. 4
    Measures, Probability and Weights

    Abstract measures, densities, and the weighted measures e^(−f) dV that Perelman uses.

    In preparation

  5. 5
    Product Measures and Change of Variables

    Fubini–Tonelli, polar coordinates, volumes of balls and the Gaussian integral.

    In preparation

  6. 6
    Modes of Convergence and Differentiation

    Covering lemmas, the maximal function and Lebesgue’s differentiation theorem.

    In preparation

  7. 7
    Lᵖ Spaces and Jensen’s Inequality

    Hölder, Minkowski, completeness, and convexity as the seed of entropy.

    In preparation

  8. 8
    Convolution and Mollifiers

    Smoothing by convolution, approximate identities, and Gaussian blur as heat flow.

    In preparation

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