Absolute continuity

In Mathematical, one can speak about absolutely continuous function and absolutely continuous measurement , and these two concepts are very dependant.

Absolutely continuous function

Motivation

A function continues

f \,

on an interval is equal to derived from its integral

\ int_a^x F (T) dt

(fundamental Theorem of the analysis). In a more general framework, that of the Integral of Lebesgue, a function

L^1 \,

Presque is equal to derived everywhere from its integral. On the other hand, a function almost everywhere derivable, even if the derivative is

L^1 \,

, can not be equal to the integral of its derivative. The staircase of the devil is an example of this pathology. The absolutely continuous functions are made to exclude this awkward phenomenon.

Definition

On $A= \,$ a Interval. It is said that the function F is absolutely continuous on has if, for any reality $\ epsilon > 0 \,$ , there exists a $\ delta > 0 \,$ such as, for any continuation $b_n_ {N} \,$ of subintervals of has disjoined interiors,

\ sum_ {N} {(b_n-a_n)} < \ delta \ Rightarrow \ sum_ {N} < \ epsilon

Properties

If a function F is continuous on a segment $\,$ , then there exists a function F integrable on $\,$ (within the meaning of Lebesgue) such as for all

x \ in, F (X) - F (a) = \ int_a^x {F (T) dt}

if and only if F is absolutely continuous on

\,

Any absolutely continuous function

on an interval with variation limited is on this interval.

If F is absolutely continuous on the interval $\,$ , then it has the property NR of Luzin: the image by $f \,$ of all Together of null measurement

(for the Measurement of Lebesgue) is of null measurement.

If F is absolutely continuous, then F is almost everywhere derivable.
If F is continuous, with limited variation and has the property NR of Luzin, then it is absolutely continuous.

Counterexample

The function continues which has as a graph the staircase of the devil is not absolutely continuous: the image of the Together of Cantor, which is of null measurement, is

\, entire

Absolutely continuous measurement

Are $\ mu$ and $\ nu$ two measurements complex on a space measured $(\ mathcal {X}, \ Tau)$ . It is said that $\ nu$ is absolutely continuous compared to $\ mu$ if and only so for any measurable unit $A$ , $\ driven (A)=0 \ naked Rightarrow \ (A)=0$ , which one notes $\ naked \ L \ mu$ .

The Théorème of Radon-Nikodym gives another characterization if $\ mu$ is positive, $\ sigma$ finished and $\ nu$ is complex, $\ sigma$ finished: there exists then $f$ measurable function such as $d \ nu=fd \ mu$ .

Bond between absolutely continuous real function and measures absolutely continuous

A measurement $\ mu$ on the whole of the real line boréliens is absolutely continuous compared to the Mesure of Lebesgue if and only if the function of associated distribution

F: X \ mapsto \ driven (] - \ infty, X])

is locally an absolutely continuous function. In other words, a function

F

is locally absolutely continuous if and only if its derived distribution is an absolutely continuous measurement compared to the measurement of Lebesgue.

See too

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