20.2 Basic Definition
The definition of the Riemannn integral of a function of n variables uses the following definition.
Definition 20.2.1 For i = 1,
k=−∞∞ be points on ℝ which satisfy
For such sequences, define a grid on ℝn denoted by G or ℱ as the collection of boxes of the
If G is a grid, ℱ is called a refinement of G if every box of G is the union of boxes of ℱ.
Lemma 20.2.2 If G and ℱ are two grids, they have a common refinement, denoted here by G∨ℱ.
be the sequences used to construct G
be the sequence
used to construct ℱ
. Now let
denote the union of
necessary to show that for each i
these points can be arranged in order. To do so, let γ0i ≡ α0i
have been chosen such that they are in order and all distinct, let γj+1i be the first element
which is larger than γji and let γ−
i be the last element of (20.3) which is strictly smaller than γ−ji.
The assumption (20.1) insures such a first and last element exists. Now let the grid G∨ℱ consist of boxes
of the form
The Riemannn integral is only defined for functions f which are bounded and are equal to zero off some
bounded set D. In what follows f will always be such a function.
Definition 20.2.3 Let f be a bounded function which equals zero off a bounded set D, and let G be a grid.
For Q ∈G, define
Also define for Q a box, the volume of Q, denoted by v
Now define upper sums, UG
and lower sums, ℒG
with respect to the indicated grid, by the
A function of n variables is Riemannn integrable when there is a unique number between all the upper and
lower sums. This number is the value of the integral.
Note that in this definition, MQ
= 0 for all but finitely many
so there are no
convergence questions to be considered here.
Lemma 20.2.4 If ℱ is a refinement of G then
Also if ℱ and G are two grids,
Proof: For P ∈G let
denote the set
Then P = ∪
Similarly, the other inequality for the upper sums is valid.
To verify the last assertion of the lemma, use Lemma 20.2.2 to write
This lemma makes it possible to define the Riemannn integral.
Definition 20.2.5 Define an upper and a lower integral as follows.
Proof: From Lemma 20.2.4 it follows for any two grids G and ℱ,
Therefore, taking the supremum for all grids on the left in this inequality,
for all grids ℱ. Taking the infimum in this inequality, yields the conclusion of the lemma. ■
Definition 20.2.7 A bounded function f which equals zero off a bounded set D, is said to be Riemannn
integrable, written as f ∈ℛ
exactly when I
. In this case define
As in the case of integration of functions of one variable, one obtains the Riemannn criterion which is
stated as the following theorem.
Theorem 20.2.8 (Riemannn criterion) f ∈ℛ
if and only if for all ε >
0 there exists a grid G such
Proof: If f ∈ℛ
and so there exist grids
Then letting ℋ = G∨ℱ, Lemma 20.2.4 implies
Conversely, if for all ε > 0 there exists G such that
Since ε > 0 is arbitrary, this proves the theorem. ■