Chapter 22: The Fundamental Theorem of Calculus

221
Vitali's theorem in **R**

Vitali's theorem for intervals in **R**.

222
Differentiating an indefinite integral

Monotonic functions are differentiable a.e., and their derivatives
are integrable; (*d/dx*)∫_{a}* ^{x}f=f* a.e.; *the
Denjoy-Young-Saks theorem.

223
Lebesgue's density theorems

*f*(*x*)=lim_{h↓0}(1/2*h*)∫_{x-h}^{x+h}*f* a.e. (*x*);
density points;
lim_{h↓0}(1/2*h*)∫_{x-h}^{x+h}|*f*-*f*(*x*)|=0 a.e.
(*x*); the Lebesgue set of a function.

224
Functions of bounded variation

Variation of a function; differences of monotonic functions; sums
and products, limits, continuity and differentiability for b.v.
functions; an inequality for ∫*f*×*g*.

225
Absolutely continuous functions

Absolute continuity of indefinite integrals; absolutely continuous
functions on **R**; integration by parts; lower semi-continuous
functions; *direct images of negligible sets; the Cantor function.

*226
The Lebesgue decomposition of a function of bounded variation

Sums over arbitrary index sets; saltus functions; the Lebesgue
decomposition.

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Revised 21.8.15