View source for Approximate limit

{{MSC|28A33|49Q15}}

[[Category:Classical measure theory]]


{{TEX|done}}

A concept of classical measure theory.

====Definition====
Consider a (Lebesgue)-measurable set $E\subset \mathbb R^n$, a measurable function $f: E\to \mathbb R$ and a point $x_0\in \mathbb R^n$ where $E$ has Lebesgue density $1$ (see [[Density of a set]]). The approximate upper and lower limits of $f$ at $x_0$ are defined, respectively, as
* The infimum of $a\in \mathbb R\cup\{\infty\}$ such that the set $\{f\leq a\}$ has density $1$ at $x_0$;
* The supremum of $a\in\{-\infty\}\cup\mathbb R$ such that the set $\{f\geq a\}$ has density $1$ at $x_0$
(cp. with Section 1.7.2 of {{Cite|EG}} and Section 12 of {{Cite|Th}}).
They are usually denoted by
\[
{\rm ap}\,\limsup_{x\to x_0}\, f(x) \qquad \mbox{and}\qquad {\rm ap}\, \liminf_{x\to x_0}\, f(x) 
\]
(some authors use also the notation $\overline{\lim}\,{\rm ap}$ and $\underline{\lim}\,{\rm ap}$). It follows from the definition that ${\rm ap}\, \liminf\leq {\rm}\, {\rm ap}\,\limsup$: if the two numbers coincide then the result is called ''approximate limit of $f$ at $x_0$'' and it is denoted by
\[
{\rm ap}\,\lim_{x\to x_0}\, f(x)\, .
\]
The approximate limit of a function taking values in a finite-dimensional vector space can be defined using its coordinate functions and the definition above.

====Properties====
Observe that the approximate limit of $f$ and $g$ are the same if $f$ and $g$ differ on a set of measure zero. A useful characterization of the approximate limit is given by the following

'''Proposition 1'''
Consider a (Lebesgue)-measurable set $E\subset \mathbb R^n$, a  measurable function $f: E\to \mathbb R$ and a point $x_0\in \mathbb R^n$. $f$ has approximate limit $L$ at $x_0$ if and only if there is a measurable set $F\subset E$ which has density $1$ at $x_0$ and such that
\[
\lim_{x\in F, x\to x_0} f(x) = L\, .
\]
In general, the existence of an ordinary limit does not follow from the  existence of an approximate limit. An approximate limit displays the  elementary properties of limits — uniqueness, and theorems on the limit  of a sum, difference, product and quotient of two functions — these properties follow indeed easily from Proposition 1.

====One-sided approximate limits====
If the domain $E$ of $f$ is a subset of $\mathbb R$ we can define  one-sided (right and left) approximate upper and lower limits: we just substitute all ''density $1$'' requirements with the ''right-hand'' or the ''left-hand density $1$'' requirement, that are, respectively,
\[
\lim_{r\downarrow 0} \frac{\lambda (G\cap ]x_0, x_0+r[)}{r} = 1 \qquad \mbox{and}\qquad \lim_{r\downarrow 0}  \frac{\lambda (G\cap ]x_0-r, x_0[)}{r} = 1\, 
\]
for a generic measurable set $G\subset \mathbb R$ (here $\lambda$ denotes the Lebesgue measure on $\mathbb R$). For instance, to define the approximate upper limit $L$ at $x_0$ of a function $f:E\to \mathbb R$ we require that the right-hand density of $E$ at $x_0$ is $1$: $L$ is then the infimum of the numbers $a\in \mathbb R\cup \{\infty\}$ such that $\{f\leq a\}$ has right-hand density $1$ at $x_0$. The corresponding notation is
\[
{\rm ap}\, \limsup_{x\to x_0^+} f(x)\, .
\]
Analogous definitions and notations hold for all the other objects.

====Comments====
Approximate limits are used to define [[Approximate  continuity|approximately continuous]] and [[Approximate  differentiability|approximate differentiable]] functions.

Approximate limits were first utilized by A. Denjoy and A.Ya. Khinchin in the study of the differential connections between an indefinite integral (in the sense of Lebesgue and in the sense of Denjoy–Khinchin).

The definitions are sometimes extended to non-measurable functions: in that case the Lebesgue measure is substituted by the Lebesgue [[Outer measure|outer measure]], see for instance Section 2.9.12 of {{Cite|Fe}} (cp. with [[Density of a set]]).

====References====
{|
|valign="top"|{{Ref|Br}}||  A.M. Bruckner,   "Differentiation of real functions" , Springer  (1978)  {{MR|0507448}}  {{ZBL|0382.26002}}
|-
|valign="top"|{{Ref|EG}}||  L.C. Evans, R.F. Gariepy,  "Measure theory  and fine properties of  functions" Studies in Advanced  Mathematics. CRC  Press, Boca Raton, FL,   1992. {{MR|1158660}}  {{ZBL|0804.2800}} 
|-
|valign="top"|{{Ref|Fe}}||    H. Federer, "Geometric measure theory". Volume 153 of Die Grundlehren    der mathematischen Wissenschaften. Springer-Verlag New York Inc., New    York, 1969. {{MR|0257325}} {{ZBL|0874.49001}}
|-
|valign="top"|{{Ref|Mu}}||   M.E. Munroe,   "Introduction to measure and integration" ,  Addison-Wesley  (1953) {{MR|035237}} {{ZBL|0227.28001}} 
|-
|valign="top"|{{Ref|Sa}}|| S. Saks,   "Theory of the integral" , Hafner  (1952) {{MR|0167578}} {{ZBL|63.0183.05}}
|-
|valign="top"|{{Ref|Th}}|| B.S. Thomson,   "Real functions" , Springer  (1985) {{MR|0818744}} {{ZBL|0581.26001}} 
|-
|}