Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

1.2. Sets 9

Example 1.2.8.LetCbe the interval of positive real numbers, i.e.,C=(0,∞).

LetAbe a subset ofC. Define the set functionQby

Q(A)=

∫

A

e−xdx, (1.2.20)

provided the integral exists. The reader should work through the following integra-
tions:

Q[(1,3)] =

∫ 3

1

e−xdx=−e−x

∣

∣

∣

∣

3

1

=e−^1 −e−^3 =0 ̇. 318

Q[(5,∞)] =

∫ 3

1

e−xdx=−e−x

∣

∣

∣

∣

∞

5

=e−^5 =0 ̇. 007

Q[(1,3)∪[3,5)] =

∫ 5

1

e−xdx=

∫ 3

1

e−xdx+

∫ 5

3

e−xdx=Q[(1,3)] +Q([3,5)]

Q(C)=

∫∞

0

e−xdx=1.

Our final example, involves anndimensional integral.

Example 1.2.9.LetC=Rn.ForAinCdefine the set function

Q(A)=

∫

···

∫

A

dx 1 dx 2 ···dxn,

provided the integral exists. For example, ifA={(x 1 ,x 2 ,...,xn):0≤x 1 ≤
x 2 , 0 ≤xi≤ 1 ,for 1 = 3, 4 ,...,n}, then upon expressing the multiple integral as
an iterated integral^3 we obtain

Q(A)=

∫ 1

0

[∫x 2

0

dx 1

]

dx 2 •

∏n

i=3

[∫ 1

0

dxi

]

=

x^22

2

∣

∣

∣

∣

1

0

• 1=
  1
  2

.

IfB={(x 1 ,x 2 ,...,xn):0≤x 1 ≤x 2 ≤···≤xn≤ 1 },then

Q(B)=

∫ 1

0

[∫xn

0

···

[∫x 3

0

[∫x 2

0

dx 1

]

dx 2

]

···dxn− 1

]

dxn

=

1

n!

,

wheren!=n(n−1)··· 3 · 2 ·1.

(^3) For a discussion of multiple integrals in terms of iterated integrals, see Chapter 3 ofMathe-
matical Comments.