29.5. THE AREA FORMULA 1019

Note there are no measurability questions in the above formula because h−1 (F) is a Borelset due to the continuity of h. The Borel measurability of J∗ (x) also follows from theobservation that h is continuous and therefore, the partial derivatives are Borel measurable,being the limit of continuous functions. Then J∗ (x) is just a continuous function of thesepartial derivatives. However, things are not so clear if F is only assumed H n measurable.Is there a similar formula for F only H n measurable?

Let λ (E) ≡ H n (E ∩h(A)) for E an arbitrary bounded H n measurable set. Thismeasure is finite on finite balls from what was shown above. Therefore, from Proposition11.7.3, there exists an Fσ set F and a Gδ set H such that F ⊆ E ⊆ H and λ (H \F) = 0.Thus

XF (h(x))J∗ (x)≤XE (h(x))J∗ (x)≤XH (h(x))J∗ (x)

where the functions on the ends are measurable. Then∫A(XH (h(x))−XF (h(x))J∗ (x))J∗ (x)dmn

= λ (H)−λ (F) = 0

and so XE (h(x))J∗ (x) = XF (h(x))J∗ (x) = XF (h(x))J∗ (x) off a set of Lebesgue mea-sure zero showing by completeness of Lebesgue measure that x→XE (h(x))J∗ (x) is Leb-esgue measurable. Then∫

h(A)XF (y)dH n =

∫AXF (h(x))J∗ (x)dmn =

∫AXE (h(x))J∗ (x)dmn

=∫

AXH (h(x))J∗ (x)dmn =

∫h(A)

XH (y)dH n

=∫

h(A)XE (y)dH n =

∫h(A)

XF (y)dH n

If E is not bounded, then replace with Er ≡ E ∩B(0,r) and pass to a limit using the mono-tone convergence theorem. This proves the following lemma.

Lemma 29.5.2 Whenever E is Lebesgue measurable,∫h(A)

XE (y)dH n =∫

AXE (h(x))J∗ (x)dmn. (29.5.23)

From this, it follows that if s is a nonnegative, H n measurable simple function, 29.5.23continues to be valid with s in place of XE . Then approximating an arbitrary nonnegativeH n measurable function g by an increasing sequence of simple functions, it follows that29.5.23 holds with g in place of XE and there are no measurability problems becausex→ g(h(x))J∗ (x) is Lebesgue measurable. This proves the following theorem which isthe area formula.

Theorem 29.5.3 Let h : Rn → Rm be Lipschitz continuous for m ≥ n. Let A ⊆ G for Gan open set be the set of x ∈ G on which Dh(x) exists, and let g : h(A)→ [0,∞] be H n

measurable. Thenx→ (g◦h)(x)J∗ (x)