Page 81 - Vector Analysis

§3.2 Properties Of The Integrals 77

3.2 Properties of the Integrals

Proposition 3.14. Let A Ď Rn be bounded, and f, g : A Ñ R be bounded. Then

ż żż ż

(a) If B Ď A, then (f 1B)(x) dx = f (x) dx and (f 1B)(x) dx = f (x) dx.

A BA B

żż ż ż żż

(b) f (x) dx + g(x) dx ď (f +g)(x) dx ď (f +g)(x) dx ď f (x) dx + g(x) dx.

AA A A AA

ż żż ż

(c) If c ě 0, then (cf )(x) dx = c f (x) dx and (cf )(x) dx = c f (x) dx. If c ă 0,

A AA A

ż żż ż

then (cf )(x) dx = c f (x) dx and (cf )(x) dx = c f (x) dx.

A AA A

żż żż

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

AA AA

ż

(e) If A has volume zero, then f is Riemann integrable over A, and f (x) dx = 0.

A

Proof. We only prove (a), (b), (c) and (e) since (d) are trivial.

(a) Let ε ą 0 be given. By the definition of the lower integral, there exist partition PA of
A and PB of B such that

ż
(f 1B)(x) dx ´ ε ă L(f 1B, PA) = ÿ inf f 1BA(x)ν(∆)

A ∆PPA xP∆

and ż ε
2
B f (x) dx ´ ă L(f, PB) = ÿ inf f B(x)ν(∆) .

∆PPB xP∆

Let PA1 be a refinement of PA such that some collection of rectangles in PA1 forms a

partition of B. Denote this partition of B by PB1 . Since inf f B(x) ď 0 if ∆ P PA1 zPB1 ,

Proposition 3.6 implies that xP∆

ż

(f 1B )(x) dx ´ ε ă L(f 1B , PA) ď L(f 1B, PA1 ) = ÿ inf f 1B A (x)ν (∆)
( )
A ∆PPA1 xP∆

= ÿÿ inf f B(x)ν(∆)
+
∆PPA1 zPB1 ∆PPB1 xP∆

ż

ď ÿ inf f B(x)ν(∆) = L(f, PB1 ) ď f (x) dx .

∆PPB1 xP∆ B