Approximate limit

2020 Mathematics Subject Classification: Primary: 28A33 Secondary: 49Q15 [MSN][ZBL]

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 [EG] and Section 12 of [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 approximately continuous and 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, see for instance Section 2.9.12 of [Fe] (cp. with Density of a set).

References

[Br]	A.M. Bruckner, "Differentiation of real functions" , Springer (1978) MR0507448 Zbl 0382.26002
[EG]	L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. MR1158660 Zbl 0804.2800
[Fe]	H. Federer, "Geometric measure theory". Volume 153 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag New York Inc., New York, 1969. MR0257325 Zbl 0874.49001
[Mu]	M.E. Munroe, "Introduction to measure and integration" , Addison-Wesley (1953) MR035237 Zbl 0227.28001
[Sa]	S. Saks, "Theory of the integral" , Hafner (1952) MR0167578 Zbl 63.0183.05
[Th]	B.S. Thomson, "Real functions" , Springer (1985) MR0818744 Zbl 0581.26001