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P. 75
§3.1 The Double Integrals 71
The two numbers
ż
f (x)dx ” inf ␣U (f, P ) ˇ P is a partition of A( ,
ˇ
A
and ż
f (x)dx ” sup ␣L(f, P ) ˇ P is a partition of A(
ˇ
A
are called the upper integral and lower integral of f over A, respective. The function
żż
f is said to be Riemann (Darboux) integrable (over A) if f (x)dx = f (x)dx,
A A
ż
and in this case, we express the upper and lower integral as f (x)dx, called the n-tuple
integral of f over A. A
Definition 3.5. A partition P1 of a bounded set A Ď Rn is said to be a refinement of
another partition P of A if for any ∆1 P P1, there is ∆ P P such that ∆1 Ď ∆. A partition
P of a bounded set A Ď Rn is said to be the common refinement of another partitions
P1, P2, ¨ ¨ ¨ , Pk of A if
1. P is a refinement of Pj for all 1 ď j ď k.
2. If P1 is a refinement of Pj for all 1 ď j ď k, then P1 is also a refinement of P.
In other words, P is a common refinement of P1, P2, ¨ ¨ ¨ , Pk if it is the coarsest refinement.
“+” “=”
Figure 3.1: The common refinement of two partitions
Qualitatively speaking, P is a common refinement of P1, P2, ¨ ¨ ¨ , Pk if for each j =
1, ¨ ¨ ¨ n, the j-th component cj of the vertex (c1, ¨ ¨ ¨ , cn) of each rectangle ∆ P P belongs to
Pi(j) for some i = 1, ¨ ¨ ¨ , k.
Proposition 3.6. Let A Ď Rn be a bounded subset, and f : A Ñ R be a bounded function.
If P and P1 are partitions of A and P1 is a refinement of P, then
L(f, P) ď L(f, P1) ď U (f, P1) ď U (f, P) .
