Metric in measure spaces - Yeh, J. (James) (New Jersey World Scientific, 2020)
Measure and metric are two fundamental concepts in measuring the size of a mathematical object. Yet there has been no systematic investigation of this relation. The book closes this gap"--
Measure and outer measure -- Construction of outer measures -- Metric outer measures and Borel outer measures -- Hausdorff measures -- Hausdorff measures on linear spaces -- Covering theorems in a metric space -- Differentiation of measures -- Averaging operators and differentiation of integrals -- Lebesgue differentiation theorem for integrals -- Hardy-Littlewood maximal functions -- Density of sets -- Approximate limit and approximate continuity -- Lipschitz mappings -- Integration with respect to a measure on a metric space.
Keywords: Mathematical analysis; Measure theory; Metric spaces