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^xf=f a.e.; *the Denjoy-Young-Saks theorem.

223 Lebesgue's density theorems
f(x)=lim_h↓0(1/2h)∫_x-h^x+hf a.e. (x); density points; lim_h↓0(1/2h)∫_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