860 APPENDIX A. THE THEORY OF THE RIEMANNN INTEGRAL∗

because∫Rm φ (·,y) dVy has the constant value given in (1.24) for x ∈ R′1. Similarly,

mR′1

(∫Rm

φ (·,y) dVy

)≡ inf

{∑

R∈Gn

(∑

P∈Gm

φ R×Pv(P)

)XR′ (x) : x ∈ R′1

}

= ∑P∈Gm

φ R1×Pv(P) . (1.25)

Theorem A.5.4 (Fubini) Let f ∈ R (Rn+m) and suppose also that f (x, ·) ∈ R (Rm) foreach x. Then ∫

Rmf (·,y) dVy ∈R (Rn) (1.26)

and ∫Rn+m

f (z) dV =∫Rn

∫Rm

f (x,y) dVy dVx. (1.27)

Proof: Let G be a grid such that UG ( f )−LG ( f )< ε and let Gn and Gm be as definedabove. Let

φ (z)≡ ∑Q∈G

MQ′ ( f )XQ′ (z) , ψ (z)≡ ∑Q∈G

mQ′ ( f )XQ′ (z) .

Observe that MQ′ ( f )≤MQ ( f ) and mQ′ ( f )≥ mQ ( f ). Then

UG ( f )≥∫

φ dV, LG ( f )≤∫

ψ dV.

Also f (z) ∈ (ψ (z) ,φ (z)) for all z. Thus from (1.24),

MR′

(∫Rm

f (·,y) dVy

)≤MR′

(∫Rm

φ (·,y) dVy

)= ∑

P∈Gm

MR′×P′ ( f )v(P)

and from (1.25),

mR′

(∫Rm

f (·,y) dVy

)≥ mR′

(∫Rm

ψ (·,y) dVy

)= ∑

P∈Gm

mR′×P′ ( f )v(P) .

Therefore,

∑R∈Gn

[MR′

(∫Rm

f (·,y) dVy

)−mR′

(∫Rm

f (·,y) dVy

)]v(R)≤

∑R∈Gn

∑P∈Gm

[MR′×P′ ( f )−mR′×P′ ( f )]v(P)v(R)≤UG ( f )−LG ( f )< ε.

This shows, from Lemma A.4.12 and the Riemannn criterion, that∫Rm f (·,y) dVy ∈R (Rn).

It remains to verify (1.27). First note∫Rn+m

f (z) dV ∈ [LG ( f ) ,UG ( f ) ] .

Next,

LG ( f )≤∫Rn+m

ψ dV =∫Rn

∫Rm

ψ dVy dVx ≤∫Rn

∫Rm

f (x,y) dVy dVx