296 CHAPTER 12. THE CONSTRUCTION OF MEASURES

unique Radon measure µ representing the functional. Thus∫R

f dµ =∫

f dF

for all f ∈Cc (R) . Now consider what this measure does to intervals. To begin with, con-sider what it does to the closed interval, [a,b] . The following picture may help.

aan

1

b bn

fn

In this picture {an} increases to a and bn decreases to b. Also suppose a,b are points ofcontinuity of F . Therefore,

F (b)−F (a)≤ L fn =∫R

fndµ ≤ F (bn)−F (an)

Passing to the limit and using the dominated convergence theorem, this shows

µ ([a,b]) = F (b)−F (a) = F (b+)−F (a−) .

Next suppose a,b are arbitrary, maybe not points of continuity of F. Then letting an and bnbe as in the above picture which are points of continuity of F,

µ ([a,b]) = limn→∞

µ ([an,bn]) = limn→∞

F (bn)−F (an)

= F (b+)−F (a−) .

In particular µ (a) = F (a+)−F (a−) and so

µ ((a,b)) = F (b+)−F (a−)− (F (a+)−F (a−))−(F (b+)−F (b−))

= F (b−)−F (a+)

This shows what µ does to intervals. This is stated as the following proposition.

Proposition 12.5.1 Let µ be the measure representing the functional

L f ≡∫

f dF, f ∈Cc (R)

for F an increasing function defined on R. Then

µ ([a,b]) = F (b+)−F (a−)

µ ((a,b)) = F (b−)−F (a+)

µ (a) = F (a+)−F (a−) .