ANOTHER PROOF OF THE CHANGE OF VARIABLE FORMULA FOR d–DIMENSIONAL INTEGRALS

Jiu Ding

The Volume 105, Number 7 issue of The American Mathematical Monthly published a new proof of the change of the variable formula for d–dimensional integrals

∫_{ℝ^{d}} f(x) dλ(x) = | det(A)| ∫_{ℝ^{d}} f(Ax) dλ(x)  (1)

with an invertible matrix A, based on group theoretical arguments [1]. In this note we provide another proof of (1) to illustrate an application of elementary measure theory and the singular value decomposition.

We use the same notation as in [1]. The measure λ ∘ A^{−1}, which is defined by (λ∘A^{−1})(B) = λ(A^{−1}(B)) for all Borel sets B, is equivalent to (i.e., absolutely continuous with respect to each other) the Lebesgue measure λ for any A ∈ GL(d, ℝ).

\underline{Proposition 1}.

∫_{ℝ^{d}} f(Ax) dλ(x) = ∫_{ℝ^{d}} f(x) d(λ ∘ A^{−1})(x).  (2)

\underline{Proof}. Let f = 1_{B}, where 1_{B} is the characteristic function of B. Then

∫_{ℝ^{d}} 1_{B}(Ax) dλ(x) = λ(A^{−1}(B)) = ∫_{ℝ^{d}} 1_{B}(x) d(λ ∘ A^{−1})(x),

i.e., (2) is true for all characteristic functions, which implies that (2) is satisfied by all simple functions. Since f is the limit of a sequence of simple functions [3], using a limiting process we see that (2) is valid for all integrable functions f.

\underline{Proposition 2}. If A ∈ GL(d, ℝ) is orthogonal, then (λ ∘ A^{−1})(B) = λ(B) for all Borel sets B.

\underline{Proof}. Every orthogonal matrix is a product of several rotations and reflections which do not change the Lebesgue measure of a Borel set.

\underline{Proposition 3}. If A ∈ GL(d, ℝ) is diagonal, then (1) is true.

\underline{Proof}. Let A = diag(a_{1}, …, a_{d}). Then it is obvious that

(λ ∘ A^{−1})(B) = | ∏_{i=1}^{d} a_{i}|^{−1}λ(B) = | det(A)|^{−1}λ(B).

Hence, (1) follows from Proposition 1.

Now, by the singular value decomposition theorem [2], A = U^{T} DV, where U and V are orthogonal matrices and D is an invertible diagonal matrix. Thus, using the above propositions,

\begin{matrix}∫_{ℝ^{d}} f(Ax) dλ(x) = ∫_{ℝ^{d}}(f ∘ U^{T} ∘ D)(V x) dλ(x) = ∫_{ℝ^{d}}(f ∘ U^{T})(Dx) dλ(x) \\ = | det(D)|^{−1} ∫_{ℝ^{d}} f(U^{T} x) dλ(x) = | det(A)|^{−1} ∫_{ℝ^{d}} f(x) dλ(x). \end{matrix}

Therefore, (1) is proved.

\underline{References}

1. P. Dierolf and V. Schmidt, “A Proof of the Change of Variable Formula for d–Dimensional Integrals,” The American Mathematical Monthly, 105 (1998), 654–656.

2. R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press, New York, 1985.

3. H. L. Royden, Real Analysis, Macmillan, New York, 1968.

Jiu Ding Department of Mathematics The University of Southern Mississippi Hattiesburg, MS 39406–5045 email: jding@yizhi.st.usm.edu