Let us denote with the -dimensional Hausdorff measure in .
A set is said to be countably -rectifiable if there exists a set which is countable -rectifiable and such that .
A set is said to be -rectifiable or simply -rectifiable if it is -rectifiable and .
If is any Borel subset of and is given, one can define the density of in as
It turns out that if is rectifiable, then in -a.e. point the density exists and is equal to . Moreover in -a.e. point there exists an approximate tangent plane to as defined below.
Given a point and a vector we say that is tangent to in if there exists a sequence of points , and a sequence of positive numbers such that
We say that a vector is approximately tangent to in if it is a tangent vector to every subset of such that . Notice that every tangent vector is also an approximately tangent vector while the converse is not always true, as it is shown in an example below. The point, here, is that being the set defined -almost everywhere, we need a stronger definition for tangent vectors.
The approximate tangent cone to in is the set of all approximately tangent vectors to in (notice that if is a tangent vector then is also a tangent vector, for all ). If the approximate tangent cone is an -dimensional vector subspace of , it is called the approximate tangent plane.
Notice that if is any -dimensional regular surface, and is the set of all points of with rational coordinates, then the set is an -rectifiable set since . Notice, however, that and consequently every vector is tangent to in every point . On the other hand the approximately tangent vectors to are only the tangent vectors to , because the set has density everywhere.
- 1 Frank Morgan: Geometric Measure Theory: A Beginner’s Guide.
|Date of creation||2013-03-22 14:28:12|
|Last modified on||2013-03-22 14:28:12|
|Last modified by||paolini (1187)|
|Defines||approximate tangent plane|