Measure, Topology, and Differentiation

R. M. Dudley

doi:10.1017/CBO9780511755347.008

7 - Measure, Topology, and Differentiation

Published online by Cambridge University Press: 06 July 2010

R. M. Dudley

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R. M. Dudley: Affiliation:
Massachusetts Institute of Technology

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Summary

Nearly every measure used in mathematics is defined on a space where there is also a topology such that the domain of the measure is either the Borel σ-algebra generated by the topology, its completion for the measure, or perhaps an intermediate σ-algebra. Defining the integrals of real-valued functions on a measure space did not involve any topology as such on the domain space, although structures on the range space ℝ (order as well as topology) were used. Section 7.1 will explore relations between measures and topologies.

The derivative of one measure with respect to another, dγ/dμ = f, is defined by the Radon-Nikodym theorem (§5.5) in case γ is absolutely continuous with respect to μ. A natural question is whether differentiation is valid in the sense that then γ(A)/μ(A) converges to f(x) as the set A shrinks down to x. For this, it is clearly not enough that the sets A contain x and their measures approach 0, as most of the sets might be far from x. One would expect that the sets should be included in neighborhoods of x forming a filter base converging to x. In ℝ, for the usual differentiation, the sets A are intervals, usually with an endpoint at x. It turns out that it is not enough for the sets A to converge to x.

Type: Chapter
Information: Real Analysis and Probability , pp. 222 - 249

DOI: https://doi.org/10.1017/CBO9780511755347.008 [Opens in a new window]

Publisher: Cambridge University Press

Print publication year: 2002

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