Page 87 - Vector Analysis

§3.4 The Fubini theorem 83

ż

Therefore, 1B(x) dx = 0, so there exists a partition P1 of R such that

R

ÿ = U (1B, P1) ă [ sup f A (R) ε ] .
ν(∆) 2 ´ inf f A(R) +1

∆PP1,∆XB‰H

()
Let U ” int ď ∆ . Then B Ď U . Since the discontinuity of f A is a subset of

∆PP1,∆XB‰H

B, f A : R X U A Ñ R is continuous. Since R X U A is closed and bounded, f A is uniformly

continuous; thus there exists δ ą 0 such that

ˇˇf A(x1) ´ f A(x2)ˇˇ ă ε if x1, x2 P R X U A and }x1 ´ x2} ă δ .
2ν(R)

Let P be a refinement of P1 such that }P} ă δ, and define two classes C1, C2 of

rectangles in P by C1 = ␣∆1 P P ˇ ∆1 Ę ∆ for all ∆ P P1 satisfying ∆XB ‰ H( and C2 =
ˇ

␣∆1 P P ˇ ∆1 R C1(. Then if ∆1 P C1, then ∆1 Ď RzU A; thus
ˇ

U (f A, P) ´ L(f A, P) = ÿ []
∆(1PP A1R)(x) A1R)(x)
sup(f ´ inf (f ν(∆1)

xP∆1 xP∆1

ÿÿ )[ ]
=+ sup f A(x) ´ inf f A(x) ν(∆1)
xP∆1
∆1PC1 ∆1PC2 xP∆1
]
ε [ ÿ
ÿ ν(∆1) + sup f A(R) ´ inf f A(R)
ď 2ν(R) ∆1PC1 ν(∆1)
ε
= ν(R) + [ ´ ] ∆1PC2
2ν(R) sup f A(R) inf f A(R)
ÿ
ν(∆)

[ ] ∆PP1 ,∆XB‰H
(R) ε
ε sup f A (R) ´ inf f A

ă 2 + [ sup f A (R) inf f A(R) + ] ă ε,
2 1
´

and we conclude that f is Riemann integrable over A by Riemann’s condition. ˝

3.4 The Fubini theorem

If f : [a, b] Ñ R is continuous, the fundamental theorem of Calculus can be applied to
computed the integral of f over [a, b]. In the following two sections, we focus on how the
integral of f over A Ď Rn, where n ě 2, can be computed if the integral exists.

Definition 3.24. Let A Ď Rn and B Ď Rm be bounded sets, S = A ˆ B be a product set in
Rn+m, and f : S Ñ R be bounded. For each fixed x P A, the lower integral of the function