Question

If \(E\) is a Borel set in \(\mathbb{R}^{n}\), the density \(D_{E}(x)\) of \(E\) at \(x\) is defined as $$ D_{E}(x)=\lim _{r \rightarrow 0} \frac{m(E \cap B(r, x))}{m(B(r, x))} $$ whenever the limit exists. a. Show that \(D_{E}(x)=1\) for a.e. \(x \in E\) and \(D_{E}(x)=0\) for a.e. \(x \in E^{c}\). b. Find examples of \(E\) and \(x\) such that \(D_{E}(x)\) is a given number \(\alpha \in(0,1)\), or such that \(D_{E}(x)\) does not exist.

Short answer

a. By the Lebesgue Density Theorem, \(D_E(x)=1\) for a.e. \(x \in E\) and \(D_E(x)=0\) for a.e. \(x \in E^c\). b. For \(\alpha \in (0,1)\), take \(E = [0, \frac{1}{2}] \cup [1, \frac{3}{2}]\). For non-existence, consider the Cantor set.

Step-by-step solution

Understanding Density Function

The density function \( D_E(x) \) measures how much of the set \( E \) is contained in an infinitesimally small ball centered at \( x \). It is defined as \[ D_E(x) = \lim_{r \to 0} \frac{m(E \cap B(r, x))}{m(B(r, x))} \], where \( B(r, x) \) is a ball of radius \( r \) centered at \( x \), and \( m(B(r, x)) \) is its Lebesgue measure.

Proving Density for Almost Every x in E

Using the Lebesgue Density Theorem, we know that for almost every point \( x \in E \), the density \( D_E(x) = 1 \). This result follows because the theorem states that for any measurable set \( E \) in \( \mathbb{R}^n \), the density is 1 at almost every point in \( E \). This implies that most of the small ball is filled by \( E \) as \( r \to 0 \).

Proving Density for Almost Every x in E^c

For the complement set \( E^c \), almost every point \( x \in E^c \) has density \( D_E(x) = 0 \). By the same theorem, the density must be either 0 or 1 for almost every \( x \), and since it is 1 in \( E \), it must be 0 in \( E^c \).

Example with Density Alpha (0,1)

Consider the set \( E = [0, \frac{1}{2}] \cup [1, \frac{3}{2}]. \) For points like \( x = \frac{1}{4} \) and \( x = \frac{5}{4}, \) the density does not have a continuous transition: small balls around these points partially lie inside and outside the intervals. For these points, \( D_E(x) \) may not be exactly 0 or 1, but 0.5, demonstrating a density value in the interval (0,1).

Example with Non-Existing Density

Take a set like the Cantor set. The density \( D_E(x) \) does not exist at points of the Cantor set since, for almost every point, the mass shares in \( \ E \) and \( \ E^c \) oscillate without converging to a specific limit. This is an example where the density limit does not exist.

Key concepts

Borel Set

In mathematics, a Borel set is any set that can be formed from open or closed sets through the operations of countable union, countable intersection, and relative complement. These sets are fundamental because they are the smallest

• σ-algebra containing the open sets in a metric space.
• They are named after the French mathematician Émile Borel.

This property makes Borel sets highly significant when dealing with measure theory, especially because any Borel set can be assigned a Lebesgue measure. Borel sets form the backbone of what we consider 'measurable' in mathematical analysis, providing a bridge between intuitive geometric concepts and abstract analytical methods.
They help in extending the notion of 'size' or 'quantity' using the Lebesgue measure, which allows for a precise way of measuring even very abstract entities like fractals or subsets of irrational numbers.

Density Function

The density function is a powerful mathematical tool used to describe how concentrated a function or a set is around a certain point. For a Borel set \( E \), the density \( D_E(x) \) at a point \( x \) is given by the formula:

• \[ D_E(x) = \lim_{r \to 0} \frac{m(E \cap B(r, x))}{m(B(r, x))} \]
• This measures the proportion of \( E \) that exists within an arbitrarily small ball centered at \( x \).

This concept is crucial in understanding how much of a set "clusters" around a given point as the ball's radius \( r \) approaches zero. When the limit of the ratio exists, it tells us whether \( x \) is a typical point of \( E \) or not. For almost every point \( x \) in \( E \), \( D_E(x) = 1 \) indicating the point is 'mostly' surrounded by \( E \), while for points in the complement set \( E^c \), the density \( D_E(x) = 0 \). This dichotomy is a direct implication of the Lebesgue Density Theorem.

Lebesgue Measure

Lebesgue measure is a fundamental mathematical concept used to assign a 'volume' or 'size' to subsets of \( \mathbb{R}^n \). While traditional notions of length, area, and volume work well for simple geometric shapes, the Lebesgue measure extends these ideas to more general sets:

• It can measure not just intervals or rectangles but also strange, complicated shapes and highly fragmented sets.
• It is defined using open covers and countably additive measures.

In essence, it's a sophisticated way of understanding size in a more flexible and inclusive manner than just piecewise geometric addition.
With the Lebesgue measure, we can successfully handle irregular sets, like those in Cantor's or Smith-Volterra-Cantor sets, providing a more accurate depiction of the 'quantity' of the set. It forms the basis for the density function and ensures that even tiny subdivisions of space are appropriately accounted for in calculations.

Complement Set

The complement set \( E^c \) refers to all the elements not in the set \( E \). If you imagine \( E \) as a region filled with points, its complement \( E^c \) comprises everything outside that region within the considered universe:

• In a pictorial sense, if \( E \) is a shaded area, \( E^c \) would be the unshaded area outside \( E \).
• This concept is essential in measure theory because it allows us to see how subsets relate to their larger 'universe' or space.

The complement set plays a vital role in defining properties like density.
The Lebesgue Density Theorem makes important statements about points in \( E^c \); for instance, for almost every point \( x \) in \( E^c \), the density \( D_E(x) \) becomes 0, highlighting that most of these points are far from the influence of \( E \). Understanding the complement of any set helps us grasp the full extent of the space and interactions that occur within it.