Measurable function Let f be a sequence of measurable functi

Measurable function

Let (f be a sequence of measurable functions defined on a measurable set E. Define Eo to Let (fn] be a sequence of measurable functions defined on a measurable set E. Define Eo to be the set of points x in E at which lfn(x)] converges. Is the set Eo measurable?

Solution

Ans-

By the Lemma on page 139,

g

n

H

=

H

i

g

n

2

H

, and

g

m

H

6

=

H

i

g

m

=

2

H

.

Therefore,

j

gH

j

=

n

in

G=H

i

n

is the smallest p ositive integer for which

g

n

2

H

. This allows you to switch b etween working in

G=H

and in

G

.

Now, consider the group

D

6

and a subgroup

Z

(

D

6

) =

f

R

0

;R

180

g

, the center

of

D

6

. That is

R

0

and

R

180

commute with any element in

D

6

. Note that

Z

(

D

6

)

is a normal subgroup of

D

6

(see Example 2 on page 179). To nd the order

of the element

R

60

Z

(

D

6

)

in

D

6

=Z

(

D

6

)

, we need to nd the smallest p ositive

integer

n

such that

R

n

60

2

Z

(

D

6

) =

f

R

0

;R

180

g

:

Let us try some values for

n

:

n

= 1

gives us

R

1

60

=

R

60

=

2

Z

(

D

6

)

.

n

= 2

gives us

R

2

60

=

R

120

=

2

Z

(

D

6

)

.

n

= 3

gives us

R

3

60

=

R

180

2

Z

(

D

6

)

.

Therefore,

n

= 3

is the smallest integer for which

R

n

60

2

Z

(

D

6

)

, and thus