1006 CHAPTER 28. HAUSDORFF MEASURE

28.4.2 Hausdorff Measure And Linear TransformationsHausdorff measure makes possible a unified development of n dimensional area. As in thecase of Lebesgue measure, the first step in this is to understand basic considerations relatedto linear transformations. Recall that for L ∈L

(Rk,Rl

),L∗ is defined by

(Lu,v) = (u,L∗v) .

Also recall Theorem 5.9.6 on Page 94 which is stated here for convenience. This theoremsays you can write a linear transformation as the composition of two linear transformations,one which preserves length and the other which distorts, the right polar decomposition.The one which distorts is the one which will have a nontrivial interaction with Hausdorffmeasure while the one which preserves lengths does not change Hausdorff measure. Theseideas are behind the following theorems and lemmas.

Theorem 28.4.3 Let F be an n×m matrix where m≥ n. Then there exists an m×n matrixR and a n×n matrix U such that

F = RU, U =U∗,

all eigenvalues of U are non negative,

U2 = F∗F, R∗R = I,

and |Rx|= |x|.

Lemma 28.4.4 Let R ∈L (Rn,Rm), n≤ m, and R∗R = I. Then if A⊆ Rn,

H n(RA) = H n(A).

In fact, if P : Rn→ Rm satisfies |Px−Py|= |x−y| , then

H n (PA) = H n (A) .

Proof: Note that

|R(x−y)|2=(R(x−y) ,R(x−y)) = (R∗R(x−y) ,x−y) = |x−y|2

Thus R preserves lengths.Now let P be an arbitrary mapping which preserves lengths and let A be bounded,

P(A)⊆ ∪∞j=1C j, r(C j)< δ , and

H nδ(PA)+ ε >

∞

∑j=1

α(n)(r(C j))n.

Since P preserves lengths, it follows P is one to one on P(Rn) and P−1 also preserveslengths on P(Rn) . Replacing each C j with C j ∩ (PA),

H nδ(PA)+ ε >

∞

∑j=1

α(n)r(C j ∩ (PA))n

=∞

∑j=1

α(n)r(P−1 (C j ∩ (PA))

)n

≥ H nδ(A).