MATH 420/507: HAUSDORFF MEASURE AND DIMENSION

For U ⊂ R^{n}, we define the diameter of U as

|U| = sup{|x − y| : x, y ∈ U}.

Let E ⊂ R^{n}, 0 < s < ∞, δ > 0. Define

H_{δ}^{s}(E) = inf ∑_{i=1}^{∞} |U_{i}|^{s},

where the infimum is taken over all countable coverings of E by open sets U_{i} of diameter < δ. Note that H_{δ_{1}}^{s}(E) ≥ H_{δ_{2}}^{s}(E) whenever δ_{1} < δ_{2} (the infimum is taken over a smaller set). Hence the limit

H^{s}(E) = lim_{δ↓0} H_{δ}^{s}(E)  (1)

exists (note that it may be infinite). It is easy to check that (1) defines an outer measure H^{s} on R^{n}, called the s-dimensional Hausdorff measure.

H^{s} is a metric outer measure: we have μ^{∗}(E ∪ F) = μ^{∗}(E) + μ^{∗}(F) for any sets E, F ⊂ X t such that dist(E, F) := inf{|x − y| : x ∈ E, y ∈ F } > 0. Consequently, all Borel sets are H^{s}-measurable. H^{s} is also a regular outer measure: for any E ⊂ R^{n} there is a G_{δ} set G such that E ⊂ G and H^{s}(E) = H^{s}(G).

Now fix δ > 0 and E ⊂ R^{n}, and let s_{1} < s_{2}. Then

H_{δ}^{s_{1}}(E) ≥ δ^{s_{1}−s_{2}}H_{δ}^{s_{2}}(E).

Taking the limit δ → 0, we see that if H^{s_{2}}(E) > 0, then H^{s_{1}}(E) = ∞. Hence there is a unique s_{0} ≥ 0 such that H^{s}(E) = 0 for s > s_{0}, and H^{s}(E) = ∞ for s > s_{0}. This value of s_{0} is called the Hausdorff dimension of E.

Example 1. Let m be the Lebesgue measure on R^{n}. Then m(E) = c_{n}H^{n}(E) for all Borel sets E, where c_{n} is the volume of an n-dimensional ball of diameter 1.

Example 2. Define a curve Γ to be the image of a continuous injection ψ : [0, 1] → R. The length of Γ may be defined as

l(Γ) = sup ∑_{i=1}^{m} |ψ(t_{i}) − ψ(t_{i−1})|  (2),

where the supremum is taken over all finite subdivisions 0 = t_{0} < t_{1} < … < t_{m−1} < t_{m} = 1 of [0, 1]. Then

H^{1}(Γ) = l(Γ).

In particular, any “fractal” curve of dimension greater than 1, such as the Koch snowflake, has infinite length.

Example 3. The 2-dimensional Hausdorff measure H^{2} may be used to define and compute the surface area of a 2-dimensional manifold (or parts thereof) in R^{3}. There is no good analogue of (2) for manifolds of dimension 2, so that in most cases it is actually necessary to use H^{2} for that purpose. Similarly for higher-dimensional submanifolds of R^{n}.

Example 4. The Cantor set C has dimension s = \frac{log 2}{log 3} = 0.6309 . . .. By modifying the construction of C (taking out intervals of length a^{n} for a ≠ 1/3), one can find subsets of R of any dimension between 0 and 1.

Let E ⊂ R^{n} be a set of Hausdorff dimension s such that H^{s}(E) > 0. The upper and lower density of E at a point x are defined as

\overline{D^{s}}(E, x) = lim sup_{r→0} \frac{H^{s}(E ∩ B(x, r))}{(2r)^{s}}, \underline{D^{s}}(E, x) = lim inf_{r→0} \frac{H^{s}(E ∩ B(x, r))}{(2r)^{s}}.

We say that a point x ∈ E is a regular point of E if \overline{D^{s}}(E, x) = \underline{D^{s}}(E, x) = 1, and irregular otherwise. E is a regular set if H^{s}-almost all its points are regular, and an irregular set if H^{s}-almost all its points are irregular.

Example 5. The Cantor set C is irregular. So is any set E of non-integer dimension (the proof of this is actually quite difficult).

Example 6. A line segment, or a smooth curve, is a set of Hausdorff dimension 1. So is the Cartesian product of two “Cantor-like” sets of dimension 1/2 (see Example 4). But a smooth curve is a regular set, while the product of two Cantor sets is irregular.