View source for Approximate differentiability

{{MSC|26B05|28A20,49Q15}}

[[Category:Classical measure theory]]


{{TEX|done}}

====Definition====
A generalization of the concept of differentiability obtained by replacing the ordinary limit by an [[Approximate limit|approximate limit]]. Consider a (Lebesgure) measurable set $E\subset \mathbb R^n$, a measurable map $f:E\to \mathbb R^k$ and a point $x_0\in E$ where $E$ has [[Density of a set|Lebesgue density]] $1$. The map $f$ is approximately differentiable at $x_0$ if there
is a linear map $A:\mathbb R^n\to \mathbb R^k$ such that
\[
{\rm ap}\, \lim_{x\to x_0} \frac{f(x)-f(x_0) - A (x-x_0)}{|x-x_0|} = 0\, ,
\]
(cp. with Section 6.1.3 of {{Cite|EG}} and Section 3.1.2 of {{Cite|Fe}}).
$A$ is then called the approximate differential of $f$ at $x_0$. If $n=1$ (i.e. $E$ is a subset of the real line), the map $A$ takes the form $A (t) = a t$: the vector $a$ is then the [[Approximate derivative|approximate derivative]] of $f$ at $x_0$, and it is sometimes denoted by $f'_{ap} (x_0)$.

====Properties====
If $f$ is approximately differentiable at $x_0$, then it is [[Approximate continuity|approximately continuous]] at $x_0$. The usual rules about uniqueness of the differential, differentiability of sums, products and quotients of functions apply to approximate differentiable functions as well and follow from a useful characterization of approximate differentiability:

'''Proposition 1'''
Consider a (Lebesgue) measurable set $E\subset \mathbb R^n$, a  measurable map $f:E\to \mathbb R^k$ and a point $x_0\in E$ where $E$ has Lebesgue density $1$. $f$ is approximately differentiable at $x_0$ if and only if there is a measurable set $F\subset E$ which has Lebesgue density $1$ at $x_0$ and such that $f|_F$ is classically differentiable at $x_0$. The approximate differential of $f$ at $x_0$ coincides then with the classical differential of $f|_F$ at $x_0$.

The chain rule applies to compositions $\varphi\circ f$ when $f$ is approximately differentiable at $x_0$ and $\varphi$ is '''classically differentiable''' at $f(x_0)$.

====Stepanov and Federer's Theorems====
The almost everywhere differentiabiliy of a function can be characterized in the following ways.

'''Theorem 2 (Stepanov)'''
A function $f:E\to\mathbb R^k$ is approximately differentiable almost everywhere if and only if the [[Approximate derivative|approximate partial derivatives]] exist almost everywhere.

For the proof see Section 3.1.4 of {{Cite|Fe}}.

'''Theorem 3 (Federer, Theorem 3.1.6 of {{Cite|Fe}})'''
Let $E\subset \mathbb R^n$ be a measurable set with finite measure. A function $f:E\to\mathbb R^k$ is approximately differentiable almost everywhere if for every $\varepsilon > 0$ there is a compact set $F\subset E$ such that $\lambda (E\setminus F)<\varepsilon$ and $f|_F$ is $C^1$ (i.e. there exists an extension $g$ of $f|_F$ to $\mathbb R^n$ which is $C^1$). 

In the latter theorem it follows also that the classical differential of $|_Ff$ coincides with the approximate differential of $f$ at almost every $x_0\in F$.

Notable examples of maps which are almost everywhere approximately differentiable are the ones belonging to the [[Sobolev classes (of functions)|Sobolev classes]] $W^{1,p}$ and to the [[Function of bounded variation|BV class]] (cp. with Theorem 4 of Section 6.1.3 of {{Cite|EG}}).

====References====
{|
|-
|valign="top"|{{Ref|AFP}}||   L. Ambrosio, N.  Fusco, D.  Pallara, "Functions of bounded variations  and  free  discontinuity  problems". Oxford Mathematical Monographs. The   Clarendon  Press,  Oxford University Press, New York, 2000.     {{MR|1857292}}{{ZBL|0957.49001}} 
|-
|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}} 
|}