Chapter 22: The Fundamental Theorem of Calculus
Vitali's theorem in R
Vitali's theorem for intervals in R.
Differentiating an indefinite integral
Monotonic functions are differentiable a.e., and their derivatives are integrable; (d/dx)∫axf=f a.e.; *the Denjoy-Young-Saks theorem.
Lebesgue's density theorems
f(x)=limh↓0(1/2h)∫x-hx+hf a.e. (x); density points; limh↓0(1/2h)∫x-hx+h|f-f(x)|=0 a.e. (x); the Lebesgue set of a function.
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.
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.
The Lebesgue decomposition of a function of bounded variation
Sums over arbitrary index sets; saltus functions; the Lebesgue decomposition.
ro-PDF (results-only version).
Return to contents page.