Chapter 26: Change of variable in the integral
Vitali's theorem in Rr
Vitali's theorem for balls in Rr; Lebesgue's Density Theorem; Lebesgue sets.
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.
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.
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.
Normalized Hausdorff measure; action of linear operators and differentiable functions; surface measure on a sphere.
The Brunn-Minkowski inequality
Arithmetic and geometric means; essential closures; the Brunn-Minkowski inequality.
ro-PDF (results-only version).
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