Absolutely continuous function

In analysis , the absolute continuity of a function is an intensification of the property of continuity . The term was introduced in 1905 by Giuseppe Vitali and allows a characterization of Lebesgue integrals .

definition

It is a finite real interval and a complex-valued function on . ${\displaystyle I\subset \mathbb {R} }$ ${\displaystyle f\colon I\to \mathbb {C} }$ ${\displaystyle I}$

The function is absolutely continuous , if for every one there, which is so small that for any finite sequence of pairwise disjoint subintervals of whose total length is valid ${\displaystyle f}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle \delta >0}$ ${\displaystyle \{]x_{k},y_{k}[\}_{1\leq k\leq n}}$ ${\displaystyle I}$ ${\displaystyle \textstyle \sum _{k=1}^{n}(y_{k}-x_{k})\,<\delta }$

${\displaystyle \sum _{k=1}^{n}\left|f(y_{k})-f(x_{k})\right|<\varepsilon .}$

Relationship to other concepts of continuity

Absolutely continuous functions are uniformly continuous and therefore especially continuous . The converse is not true, so the Cantor function is continuous, but not absolutely continuous. On the other hand, every Lipschitz continuous function is also absolutely continuous.

Absolute continuity of dimensions

The real-valued, absolutely continuous functions are of particular importance for measure theory . It denotes the Lebesgue measure . For monotonically increasing real-valued functions , the following properties are equivalent: ${\displaystyle \lambda }$ ${\displaystyle f\colon I=[a,b]\to \mathbb {R} }$

1. The function is absolutely steady on ${\displaystyle f}$ ${\displaystyle I}$ .
2. The function forms - zero amounts to zero sets off, ie for all measurable amounts applies ${\displaystyle f}$ ${\displaystyle \lambda }$ ${\displaystyle A\subseteq I}$ ${\displaystyle \lambda (A)=0\implies \lambda (f(A))=0}$ .
3. The function is - almost everywhere differentiable , the derivative function is integrated and for all true . ${\displaystyle f}$ ${\displaystyle \lambda }$ ${\displaystyle f'\in L^{1}(\lambda )}$ ${\displaystyle x\in I}$ ${\displaystyle \textstyle f(x)-f(a)=\int _{a}^{x}f'(t)\ \mathrm {d} \lambda (t)}$

This results in a close connection between the absolutely continuous functions and the absolutely continuous measures , which is conveyed by the distribution functions .

A measure is absolutely continuous with respect to if and only if every restriction of the distribution function from to a finite interval is an absolutely continuous function from . ${\displaystyle \mu }$ ${\displaystyle \lambda }$ ${\displaystyle \mu }$ ${\displaystyle I\subset \mathbb {R} }$ ${\displaystyle I}$

Lebesgue integrals

The absolutely continuous functions are also used in integration theory , where they serve to extend the fundamental theorem of analysis to Lebesgue integrals. Beyond the above equivalence, non-monotonic, absolutely continuous functions are also differentiable almost everywhere and it is true . In addition, it is weakly differentiable and the weak derivative agrees (almost everywhere) with . This actually provides a characterization of the Lebesgue integrability, because the following inversion also applies to any functions: ${\displaystyle \textstyle f(x)-f(a)=\int _{a}^{x}f'\,\mathrm {d} \lambda }$ ${\displaystyle f}$ ${\displaystyle f'}$

If a function has an integrable derivation function and if that applies to all , then it is necessary to be absolutely continuous . ${\displaystyle f\colon I=[a,b]\to \mathbb {R} }$ ${\displaystyle f'\in L^{1}}$ ${\displaystyle x\in I}$ ${\displaystyle \textstyle f(x)-f(a)=\int _{a}^{x}f'(t)\ \mathrm {d} \lambda (t)}$ ${\displaystyle f}$ ${\displaystyle I}$

Optimal control

In the theory of optimal controls it is required that the solution trajectories are absolutely continuous.

literature

• Jürgen Elstrodt : Measure and integration theory. 4th, corrected edition. Springer, Berlin 2005, ISBN 3-540-21390-2 .
• Walter Rudin : Real and Complex Analysis . 3. Edition. McGraw-Hill, New York 1987 (English).

Individual evidence

1. ↑ Giuseppe Vitali: Opere sull'analisi real e complessa. Edizioni Cremonese, Bologna 1984
2. ↑ Jürgen Elstrodt: Measure and integration theory. 4th, corrected edition. Springer, Berlin 2005, ISBN 3-540-21390-2 , p. 281.