Page 85 - Vector Analysis

§3.3 Integrability for Continuous Functions 81

Theorem 3.17. Let A Ď Rn be bounded, c P R, and f, g : A Ñ R be Riemann integrable.
Then

ż żż

(a) f ˘ g is Riemann integrable, and (f ˘ g)(x) dx = f (x) dx ˘ g(x) dx.

A AA

żż

(b) cf is Riemann integrable, and (cf )(x) dx = c f (x) dx.

AA

ˇż ˇ ż
(c) |f | is Riemann integrable, and ˇ f (x) dxˇ ď |f (x)|dx.

ˇˇ

AA

żż

(d) If f ď g, then f (x) dx ď g(x) dx.

AA

ˇż ˇ
(e) If A has volume and |f | ď M , then ˇ f (x) dxˇ ď M ν(A).

ˇˇ

A

Definition 3.18. Let A Ď Rn be a set and f : A Ñ R be a function. For B Ď A, the

restriction of f to B is the function f ˇ : A Ñ R given by f |B = f 1B. In other words,
ˇB

" f (x) if x P B ,
0 if x P AzB .
f ˇ (x) =
ˇB

The following two theorems are direct consequences of (a) of Proposition 3.14 and Corol-
lary 3.16.

Theorem 3.19. Let A, B Ď Rn be bounded, B Ď A, and f : A Ñ R be a bounded function.

Then f is Riemann integrable over B if and only if f |B is Riemann integrable over A. In

either cases,

żż

f ˇ (x) dx = f (x) dx .
ˇB
AB

Theorem 3.20. Let A, B be bounded subsets of Rn be such that A X B has volume zero, and

f : AYB Ñ R be bounded such that f ˇ and f ˇ are al l Riemann integrable over A Y B.
ˇA ˇB

Then f is Riemann integrable over A Y B, and

ż żż

f (x) dx = f (x) dx + f (x) dx .

AYB AB

3.3 Integrability for Almost Continuous Functions

Lemma 3.21. Let A Ď Rn be a bounded set of volume zero. If B Ď A, then B has volume
zero.