Chapter 26: Change of variable in the integral

261
Vitali's theorem in **R**^{r}

Vitali's theorem for balls in **R**^{r}; Lebesgue's Density Theorem;
Lebesgue sets.

262
Lipschitz and differentiable functions

Lipschitz functions; elementary properties; differentiable
functions from **R**^{r} to **R**^{s}; differentiability and partial
derivatives; approximating a differentiable function by piecewise
affine functions; *Rademacher's theorem.

263
Differentiable transformations in **R**^{r}

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 **R**^{s}; 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