Contents of Measure Theory, by D.H.Fremlin

Chapter 26: Change of variable in the integral

261 Vitali's theorem in Rr
Vitali's theorem for balls in Rr; Lebesgue's Density Theorem; Lebesgue sets.

262 Lipschitz and differentiable functions
Lipschitz functions; elementary properties; differentiable functions from Rr to Rs; differentiability and partial derivatives; approximating a differentiable function by piecewise affine functions; *Rademacher's theorem.

263 Differentiable transformations in Rr
In the formula ∫g(y)dy=∫J(x)g(φ(x))dx, find J when φ is (i) linear (ii) differentiable; detailed conditions of applicability; polar coordinates; the case of non-injective φ; the one-dimensional case.

264 Hausdorff measures
r-dimensional Hausdorff measure on Rs; Borel sets are measurable; Lipschitz functions; if s=r, we have a multiple of Lebesgue measure; *Cantor measure as a Hausdorff measure.

265 Surface measures
Normalized Hausdorff measure; action of linear operators and differentiable functions; surface measure on a sphere.

*266 The Brunn-Minkowski inequality
Arithmetic and geometric means; essential closures; the Brunn-Minkowski inequality.

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Revised 18.8.15