The definition of the Riemannn integral of a function of n variables uses the following definition.
Definition C.2.1 For i = 1,
 (3.1) 
For such sequences, define a grid on ℝ^{n} denoted by G or ℱ as the collection of boxes of the form
 (3.2) 
If G is a grid, ℱ is called a refinement of G if every box of G is the union of boxes of ℱ.
Proof: Let

have been chosen such that they are in order and all distinct, let γ_{j+1}^{i} be the first element of
 (3.3) 
which is larger than γ_{j}^{i} and let γ_{−}

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 C.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
 (3.4) 
Also define for Q a box, the volume of Q, denoted by v


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, M_{Q}
Proof: For P ∈G let

Then P = ∪


Similarly, the other inequality for the upper sums is valid.
To verify the last assertion of the lemma, use Lemma C.2.2 to write

This lemma makes it possible to define the Riemannn integral.
Lemma C.2.6 I
Proof: From Lemma C.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 C.2.7 A bounded function f which equals zero off a bounded set D, is said to be Riemannn integrable, written as f ∈ℛ

As in the case of integration of functions of one variable, one obtains the Riemannn criterion which is stated as the following theorem.
Theorem C.2.8 (Riemannn criterion) f ∈ℛ

Proof: If f ∈ℛ

Then letting ℋ = G∨ℱ, Lemma C.2.4 implies

Conversely, if for all ε > 0 there exists G such that

then

Since ε > 0 is arbitrary, this proves the theorem. ■