858 APPENDIX A. THE THEORY OF THE RIEMANNN INTEGRAL∗

is a function of y for each x ∈Rn. Therefore, it might be possible to integrate this functionof y and write ∫

Rmf (x,y) dVy.

Now the result is clearly a function of x and so, it might be possible to integrate this andwrite ∫

Rn

∫Rm

f (x,y) dVy dVx.

This symbol is called an iterated integral, because it involves the iteration of two lowerdimensional integrations. Under what conditions are the two iterated integrals equal to theintegral ∫

Rn+mf (z) dV ?

Definition A.5.1 Let G be a grid on Rn+m defined by the n+m sequences,{α

ik}∞

k=−∞i = 1, · · · ,n+m.

Let Gn be the grid on Rn obtained by considering only the first n of these sequences andlet Gm be the grid on Rm obtained by considering only the last m of the sequences. Thus atypical box in Gm would be

n+m

∏i=n+1

[α

iki,α i

ki+1], ki ≥ n+1

and a box in Gn would be of the form

n

∏i=1

[α

iki,α i

ki+1], ki ≤ n.

Lemma A.5.2 Let G , Gn, and Gmbe the grids defined above. Then

G = {R×P : R ∈ Gn and P ∈ Gm} .

Proof: If Q∈ G , then Q is clearly of this form. On the other hand, if R×P is one of thesets described above, then from the above description of R and P, it follows R×P is one ofthe sets of G . ■

Now let G be a grid on Rn+m and suppose

φ (z) = ∑Q∈G

φ QXQ′ (z) (1.18)

where φ Q equals zero for all but finitely many Q. Thus φ is a step function. Recall that for

Q =n+m

∏i=1

[ai,bi] , Q′ ≡n+m

∏i=1

(ai,bi]

The functionφ = ∑

Q∈Gφ QXQ′