# Approximate limit

2020 Mathematics Subject Classification: Primary: 26B05 Secondary: 28A2049Q15 [MSN][ZBL]

A concept of classical measure theory.

## Contents

#### 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.