It was shown in Lemma 26.4.5 that

where F = RU with R preserving distances and U a symmetric matrix having all positive eigenvalues. The area formula gives a generalization of this simple relationship to the case where F is replaced by a nonlinear mapping h. It contains as a special case the earlier change of variables formula. There are two parts to this development. The first part is to generalize Lemma 26.4.5 to the case of nonlinear maps. When this is done, the area formula can be presented.
In the first version of the area formula h will be a Lipschitz function,

defined on ℝ^{n} which is one to one on G, a measurable subset of ℝ^{n}. This is no loss of generality because of Theorem 24.5.4.
The following lemma states that Lipschitz maps take sets of measure zero to sets of measure zero. It also gives a convenient estimate. It involves the consideration of ℋ^{n} as an outer measure. Thus it is not necessary to know the set B is measurable.
Lemma 27.0.1 If h is Lipschitz with Lipschitz constant K then

Also, if T is a set in ℝ^{n}, m_{n}
Proof: Let

Then

Now take a limit as δ → 0. The second claim follows from m_{n} = ℋ^{n} on Lebesgue measurable sets of ℝ^{n}. ■
Lemma 27.0.2 If S is a Lebesgue measurable set and h is Lipschitz then h

It is not necessary that h be one to one.
Proof: The estimate follows from Lemma 27.0.1 and the observation that, as shown before, Theorem 26.2.3, if S is Lebesgue measurable in ℝ^{n}, then ℋ^{n}

The second set is Borel and the first has ℋ^{n} measure zero. By completeness of ℋ^{n}, h
By Theorem 4.13.6 on Page 262, when Dh

where
Proof: First note that
Then the following corollary follows from Lemma 27.0.3.
a decomposition
First is a simple lemma which is fairly interesting.
Lemma 27.0.5 Suppose S,T are linear, S^{−1} exists and let δ ∈
 (27.0.1) 
for all v≠0. Similarly if T^{−1} exists and

Proof: Suppose for some v≠0,

Hence, since

and so one would need to have
If


Thus, as before, δ ≤

if v≠0. The last assertion follows by noting that if T^{−1} is given to exist and S is close to T then

By choosing δ appropriately, one can achieve the last inclusion for given
In short, the above lemma says that if one of S,T is invertible and the other is close to it, then it is also invertible and the quotient of
Lemma 27.0.6 Let S,T be n × n matrices which are invertible. Then

and if L is a continuous linear transformation such that for a < b,

If
Proof: Consider the first claim. For

Thus o
Consider the second claim. Pick δ small. Then by Lemma 27.0.5

if δ is small enough. The other inequality is shown exactly similar. ■
The following is a simplified version of an argument in [?]. Assume the following:
 (27.0.2) 
By regularity, we can and will assume A is a Borel set. Of course this is automatic if h is Lipschitz which is the case of interest here but the condition 27.0.2 could likely be obtained in other situations also.
For x ∈ A, let Dh
Let B be a Lebesgue measurable subset of A^{+} and let b ∈ B. Let S be a countable dense subset of the space of symmetric invertible matrices and let C be a countable dense subset of B.
Let ε be a small number. Since U
 (27.0.3) 
provided that a ∈ B
 (27.0.4) 
where U

 (27.0.5) 
Thus there are countably many of these sets E
For a,b ∈ E
 (27.0.6) 
and so

Note that this proves that on E
Lemma 27.0.7 For x ∈ A^{+} the set where U
It follows from this that
 (27.0.7) 
 (27.0.8) 
Here the variables are in the appropriate sets, T
Now let
Lemma 27.0.8 There are disjoint measurable sets E_{k} whose union equals B and symmetric linear transformations T_{k} such that
 (27.0.9) 
 (27.0.10) 
on T_{k}
 (27.0.11) 
One can also conclude that for b ∈ E_{k},
 (27.0.12) 
Proof: It only remains to verify the last claim. However, this follows right away from 27.0.11. This formula implies that

and so

This implies

and so

This lemma, along with Lemma 27.0.1 about the relationship between Hausdorff measure and Lipschitz mappings, implies the following.
