The most Striking Theorem in Real Analysis

Lebesgue’s Theorem (see below) has been called one of the most striking theorems in real analysis. Indeed it is a very surprising result.

Lebesgue’s Theorem (Monotone functions)

If the function f is monotone on the open interval (a,b), then it is differentiable almost everywhere on (a,b).

Absolutely Continuous Functions


A real-valued function f on a closed, bounded interval [a,b] is said to be absolutely continuous on [a,b] provided for each \epsilon>0, there is a \delta>0 such that for every finite disjoint collection \{(a_k,b_k)\}_{k=1}^n of open intervals in (a,b), if \displaystyle \sum_{k=1}^n(b_k-a_k)<\delta, then \displaystyle \sum_{k=1}^n|f(b_k)-f(a_k)|<\epsilon.

Equivalent Conditions

The following conditions on a real-valued function f on a compact interval [a,b] are equivalent:
(i) f is absolutely continuous;

(ii) f has a derivative f' almost everywhere, the derivative is Lebesgue integrable, and \displaystyle f(x)=f(a)+\int_a^x f'(t)\,dt for all x on [a,b];

(iii) there exists a Lebesgue integrable function g on [a,b] such that \displaystyle f(x)=f(a)+\int_a^x g(t)\,dt for all x on [a,b].

Equivalence between (i) and (iii) is known as the Fundamental Theorem of Lebesgue integral calculus.

Author: mathtuition88

Math and Education Blog

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