320 CHAPTER 11. REGULAR MEASURES

≤ |t0|

µ(spt( f ))︷ ︸︸ ︷n

∑i=1

µ(Ei)+ε

nn |t0|+∑

itiµ (Ei)+∑

iti

ε

n+∑

iεµ (Ei)+

ε2

n−|t0|µ(spt( f ))

≤ ε |t0|+ ε (|t0|+ |b|)+ εµ(spt( f ))+ ε2 +∑

itiµ (Ei)

≤ ε |t0|+ ε (|t0|+ |b|)+2εµ(spt( f ))+ ε2 +

n

∑i=1

ti−1µ(Ei)

≤ ε (2 |t0|+ |b|+2µ(spt( f ))+ ε)+∫

f dµ

Since ε > 0 is arbitrary, L f ≤∫

f dµ for all f ∈Cc(X), f real. Hence equality holds becauseL(− f ) ≤ −

∫f dµ so L( f ) ≥

∫f dµ . Thus L f =

∫f dµ for all f ∈ Cc(X). Just apply the

result for real functions to the real and imaginary parts of f . ■Using Corollary 9.8.9 we obtain the following corollary. Note that the conditions of

the above theorem imply that X is a Polish space in the usual case where closed balls arecompact.

Corollary 11.2.3 If X is a Polish space then in the above theorem, we obtain innerregularity of µ in terms of compact sets. That is if F ∈F , then

µ (F) = sup{µ (K) : K ⊆ F,K compact}

11.3 The p Dimensional Lebesgue MeasureTheorem 11.2.2 will provide many examples of Radon measures on Rp. Lebesgue measureis obtained by letting

L f ≡∫R· · ·∫R

f (x1, ...,xp)dm1 (x1) · · ·dmp (xp)

for f ∈Cc (Rp). Thus Lebesgue measure is a Radon measure, denoted as mp. In this case,the σ algebra will be denoted as Fp. Lebesgue measure also has other very importantproperties. Integrals can be easily computed and the measure is translation invariant.

Theorem 11.3.1 Whenever f is measurable and nonnegative, then whenever g isBorel measurable and equals f a.e. and h is Borel and equals f a.e.∫

R· · ·∫R

h(x1, ...,xp)dm1 (xi1) · · ·dmp(xip

)=

∫Rp

f dmp =∫R· · ·∫R

g(x1, ...,xp)dm1 (xi1) · · ·dmp(xip

)where (i1, i2, ..., ip) is any permutation of the integers {1,2, ..., p}. Also, mp is regular andcomplete. If R is of the form ∏

pi=1 Ii where Ii is an interval, then mp (R) is the product of the

lengths of the sides of R. Also if E ∈Fp, then mp (x+E) = mp (E).