12.4. ONE DIMENSIONAL LEBESGUE MEASURE 295

12.4 One Dimensional Lebesgue MeasureTo obtain one dimensional Lebesgue measure, you use the positive linear functional L givenby

L f =∫

f (x)dx

whenever f ∈Cc (R) . Lebesgue measure, denoted by m is the measure obtained from theRiesz representation theorem such that∫

f dm = L f =∫

f (x)dx.

From this it is easy to verify that

m([a,b]) = m((a,b)) = b−a. (12.4.16)

This will be done in general a little later but for now, consider the following picture offunctions, f k and gk. Note that f k ≤X(a,b) ≤X[a,b] ≤ gk.

a+1/k

a

1b−1/k

b

f k

a−1/k

a

1

b

b+1/kgk

Then considering lower sums and upper sums in the inequalities on the ends,(b−a− 2

k

)≤

∫f kdx =

∫f kdm≤ m((a,b))≤ m([a,b])

=∫

X[a,b]dm≤∫

gkdm =∫

gkdx≤(

b−a+2k

).

From this the claim in 12.4.16 follows.

12.5 One Dimensional Lebesgue Stieltjes MeasureThis is just a generalization of Lebesgue measure. Instead of the functional,

L f ≡∫

f (x)dx, f ∈Cc (R) ,

you use the functional

L f ≡∫

f (x)dF (x) f ∈Cc (R) ,

where F is an increasing function defined on R. By Theorem 4.3.4 this functional is easilyseen to be well defined. Therefore, by the Riesz representation theorem there exists a