11.3. THE p DIMENSIONAL LEBESGUE MEASURE 321

Proof: Let K consist of all open rectangles ∏i (ai,bi) along with /0 and Rp. Thus thisis a π system. Let Rn ≡ ∏

pi=1 (−n,n). Let G consist of the Borel sets E ⊆ Rp such that∫

Rn∩E dmp =∫R · · ·

∫RXRn∩Edm1 (xi1) · · ·dmp

(xip

). Then K ⊆ G . It is obvious from the

monotone convergence theorem that G is closed with respect to countable disjoint unions.Indeed, this theorem implies that for E = ∪iEi, the Ei disjoint,

∫Rn∩E

dmp = limm→∞

∫Rn∩∪m

i=1Ei

dmp = limm→∞

(m

∑i=1

∫XRn∩Eidmp

)

= limm→∞

(m

∑i=1

∫R· · ·∫R

XRn∩Eidm1 (xi1) · · ·dmp(xip

))

= limm→∞

(∫R· · ·∫R

m

∑i=1

XRn∩Eidm1 (xi1) · · ·dmp(xip

))

= limm→∞

(∫R· · ·∫R

XRn∩∪mi=1Eidm1 (xi1) · · ·dmp

(xip

))=

(∫R· · ·∫R

XRn∩Edm1 (xi1) · · ·dmp(xip

))As to complements,

∫Rn

dmp =∫

Rn∩EC dmp +∫

Rn∩E dmp. Thus∫R· · ·∫R

XRn∩EC dm1 (xi1) · · ·dmp(xip

)=

∫R· · ·∫R(XRn −XRn∩E)dm1 (xi1) · · ·dmp

(xip

)=∫

Rn∩ECdmp

It follows that G = B (Rp) , the Borel sets. Hence∫Rp

XEdmp =∫R· · ·∫R

XEdm1 (xi1) · · ·dmp(xip

)for any Borel set E after letting n→ ∞ and using the monotone convergence theorem.Approximating a nonnegative Borel function g with an increasing sequence of simple Borelmeasurable functions, and using the monotone convergence theorem yields∫

gdmp =∫R· · ·∫R

g(x1, ...,xp)dm1 (xi1) · · ·dmp(xip

)The claim about the measure of a box being the product of the lengths of its sides alsocomes from this.

By Proposition 11.1.2, for f measurable, there exists g Borel measurable such thatg = f a.e. and g≤ f . Then∫

Rpf dmp =

∫Rp

gdmp =∫R· · ·∫R

g(x1, ...,xp)dm1 (xi1) · · ·dmp(xip

)It is similar if h≥ f and equal to f a.e.

It remains to consider the claim about translation invariance. If R is a box, R =

∏pi=1 (ai,bi) , then it is clear that mp (x+R) = mp (R). Let K be as above and let G be

those Borel sets E for which mp (x+E ∩Rn) = mp (E ∩Rn) where Rn is as above. Thus G