Page 82 - Vector Analysis
P. 82
78 CHAPTER 3. Multiple Integrals
On the other hand, let PrA be a partition of A such that PB Ď PrA and
ÿε
ν(∆) ď 2(M + 1) ,
∆PPrAzPB , ∆XB‰H
where M ą 0 is an upper bound of |f |. Then
ÿ inf f B(x)ν(∆) ě ´M ÿ ν(∆) ě ´ ε
xP∆ 2
∆PPrAzPB , ∆XB‰H ∆PPrAzPB , ∆XB‰H
which further implies that
ż
(f 1B)(x) dx ě L(f 1B, PrA) = ÿ inf f 1BA(x)ν(∆)
xP∆
A
( ∆PPrA
ÿÿ ÿ ) inf f B(x)ν(∆)
=+ + xP∆
∆PPrAzPB , ∆XB=H
∆PPB ∆PPrAzPB , ∆XB‰H
ż
= L(f, PB) + ÿ inf f B(x)ν(∆) ą f (x) dx ´ ε .
xP∆
∆PPrAzPB , ∆XB‰H B
Therefore, we establish that
żżż
f (x) dx ´ ε ă (f 1B)(x) dx ă f (x) dx + ε .
BAB
żż
Since ε ą 0 is given arbitrarily, we conclude that (f 1B)(x) dx = f (x) dx. Similar
AB
żż
argument can be applied to conclude that (f 1B)(x) dx = f (x) dx.
AB
(b) Let ε ą 0 be given. By the definition of the lower integral, there exist partitions P1
and P2 of A such that
ż f (x) dx ´ ε ă L(f, P1) and ż ε ă L(g, P2) .
2 2
A g(x) dx ´
A
Let P be a common refinement of P1 and P2. Then
żż
f (x) dx + g(x) dx ´ ε ă L(f, P1) + L(f, P2) ď L(f, P) + L(g, P)
AA
= ÿ inf f (x)ν(∆) + ÿ inf g(x)ν(∆)
∆PP xP∆ ∆PP xP∆
ż
ď ÿ inf (f + g)(x)ν(∆) = L(f + g, P) ď (f + g)(x) dx .
xP∆
∆PP A
