9.5 Change Of Variables, Linear Maps
The change of variables formula for linear maps is implied by the following major theorem.
First, here is some notation. Let dxi denote
First of all is an important observation about measurability.
Definition 9.5.1 A function h : ℝp → ℝp is said to be a Lipschitz function
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 mp
, it follows that mp
. Also if E ∈ℱp, then h
Proof: By the Lipschitz condition,
and you can simply let
T ⊆ V
Then there is a countable disjoint sequence of balls
is contained in V
. Then h
Since ε is arbitrary, this shows that h
is measurable and
Now let E ∈ℱp, mp
. Then by regularity, there exists F
which is the countable
union of compact sets such that
where N is a set of measure zero. Then from the first part, h
and this set on
the right has measure zero and so by completeness of the measure,
because F = ∪kKk, each Kk compact. Hence h
which is the countable union
of compact sets due to the continuity of
. For arbitrary E, 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 mp
. Also, if E is any Lebesgue
measurable set, then
Proof: Let Q denote ∏
an open rectangle. First suppose A
is an elementary
matrix which is obtained from I
by adding α
times row i
. Then det
= 1 and
The linear transformation determined by A
just shears Q
in the direction of xj
. It is clear in
this case that AQ
is an open, hence measurable set. A picture of the effect of doing A
follows in case α
Next suppose A is an elementary matrix which comes from multiplying the jth row of I
with α≠0. Then this changes the length of one of the sides of the box by the factor α,
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,
= 1 and
Let Rn = ∏
consist of all finite intersections of such open boxes as
just described. This is clearly a π
system. Now let
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 EC ∈G? Since A is onto, AEC = ℝn ∖ AE which is
Borel. What about the estimate?
It was shown above that G
. By Lemma 7.4.2
, it follows that G⊇ σ
Therefore, for any
elementary and E
a Borel set,
Now consider A an arbitrary invertible matrix. Then A is the product of elementary
and so if E
is Borel, so is AE
In case A is an arbitrary matrix which has rank less than p, there exists a sequence of
elementary matrices E1,E2,
where B is in row reduced echelon form and has at least one row of zeros. Thus if S is any
Lebesgue measurable set,
is a measurable set of measure zero because it is contained in a set of the
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 Gδ sets to
Gδ sets and Fσ sets to Fσ sets. Let E be Lebesgue measurable. By regularity of the measure,
there exists G and F, Gδ and Fσ sets respectively such that F ⊆ E ⊆ G and mp
AF ⊆ AE ⊆ AG
By completeness, AE is Lebesgue measurable. Also
The above theorem also implies easily the following version of the change of variables formula
for linear mappings.
Theorem 9.5.4 Let f ≥ 0 and suppose it is Lebesgue measurable. Then if A is a
p × p matrix,
Proof: From Theorem 9.5.3, the equation is true if det
= 0. It follows that it suffices
to consider only the case where
exists. First suppose f
is a Lebesgue measurable set. In this case, A
Then from Theorem
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, mp
because the linear transformation
x → αx
has determinant equal to αp
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