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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