The change of variables formula for linear maps is implied by the following major theorem. First, here is some notation. Let dx_{i} denote

First of all is an important observation about measurability.
Definition 9.5.1 A function h : ℝ^{p} → ℝ^{p} is said to be a Lipschitz function if

for some (any norm).
The nice thing about Lipschitz functions is that they take Lebesgue measurable sets to Lebesgue measurable sets.
Theorem 9.5.2 In case h is Lipschitz,

then if m_{p}
Proof: By the Lipschitz condition,

Then there is a countable disjoint sequence of balls

Since ε is arbitrary, this shows that h
Now let E ∈ℱ_{p}, m_{p}

where N is a set of measure zero. Then from the first part, h

because F = ∪_{k}K_{k}, each K_{k} compact. Hence h
Of course an example of a Lipschitz map is a linear map.

Therefore, if A is linear and E is Lebesgue measurable, then AE is also Lebesgue measurable. This is convenient.
Theorem 9.5.3 Let E be any Lebesgue measurable set and let A be a p×p matrix. Then AE is Lebesgue measurable and m_{p}

Proof: Let Q denote ∏ _{i=1}^{p}
Next suppose A is an elementary matrix which comes from multiplying the j^{th} row of I with α≠0. Then this changes the length of one of the sides of the box by the factor α, resulting in

In case A is an elementary matrix which comes from switching two rows of the identity, then it maps Q to a possibly different Q which has the lengths of its sides the same set of numbers as the set of lengths of the sides of Q. Therefore,

Let R_{n} = ∏ _{i=1}^{p}

where A is any of the above elementary matrices. It is clear that G is closed with respect to countable disjoint unions. If E ∈G, is E^{C} ∈G? Since A is onto, AE^{C} = ℝ^{n} ∖ AE which is Borel. What about the estimate?

and so
Now consider A an arbitrary invertible matrix. Then A is the product of elementary matrices A_{1}


In case A is an arbitrary matrix which has rank less than p, there exists a sequence of elementary matrices E_{1},E_{2},

where B is in row reduced echelon form and has at least one row of zeros. Thus if S is any Lebesgue measurable set,

and so

and this has measure zero.
It has now been shown that for invertible A, and E any Borel set,

and for any Lebesgue measurable set E and A not invertible, the above formula holds. It only remains to verify the formula holds for A invertible and E only Lebesgue measurable. However, in this case, A maps open sets to open sets because its inverse is continuous and maps compact sets to compact sets because x → Ax is continuous. Hence A takes G_{δ} sets to G_{δ} sets and F_{σ} sets to F_{σ} sets. Let E be Lebesgue measurable. By regularity of the measure, there exists G and F, G_{δ} and F_{σ} sets respectively such that F ⊆ E ⊆ G and m_{p}

By completeness, AE is Lebesgue measurable. Also
Proof: From Theorem 9.5.3, the equation is true if det


It follows from this that 9.8 holds whenever f is a nonnegative simple function. Finally, the general result follows from approximating the Lebesgue measurable function with nonnegative simple functions using Theorem 7.1.6 and then applying the monotone convergence theorem. ■
Note that it now follows that for the ball defined by any norm, m_{p}
This is now a very good change of variables formula for a linear transformation. Next this is extended to differentiable functions.
class=”left” align=”middle”(Y )9.6. DIFFERENTIABLE FUNCTIONS AND MEASURABILITY