Analysis 1 (MATS3103)
Mathematics and Computer Science - MCS
Semester: First Semester
Level: 200
Year: 2016
REPUBLIC O F CA M EROON
Peace-Work-Fatherland
MINIS TRY OF HIGHER
EDUCATION
REPUBLIQUE DU CAMERROUN
Paix-Travaille-Patrie
MLNISTERE DE L'ENSEIGNEIMIENT
SUPERIEURE
THE UNIVERSITY OF BAMENDA UNIVERSITE DE BAM.ENDA
FACULTY OF SCIENCE
P.O Box 39 BA.MBILI
TEL: (+23) 22 81 63 54
FA C U LTE D E S C IEN CE
TUTORIAL SHEET 3 2015/2016
DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE
COURSE CODE: .MATS2101 SHU F. CHE
Answer all questions. Assignment will be submitted in 14 days counted from November 4, 2016
1a. Let Ω be a set and be the power set of Ω. Let x, y Ω be such that x y. Define
µ on by µ(A) =
Show that µ is not a content of
b. Let Ω be a countable set and be the subsets of Ω of finite cardinality. For each
A , define µ(A):=d(A), the cardinality of A. Which of the following statements
are true?
i) is a ring
ii) is an algebra
iii) is not a –algebra
iv) µ is a constant on
v) µ is a pre-measure of
vi) µ is a measure of
2a. Let (Ω, , µ) be a measure space. Suppose that µ is -finite and µ(Ω) - +∞. Show
that for every m , there exist a set A such that ∞
2b. Would the statement in (a) remain true if µ is finite
3. Let Ω be an infinite set and be the power set of Ω. Define µ on by
µ(A) =
!
Show that µis a content on but not a pre-measure
4. Let (Ω, ) be a measurable space and µ
1
,…, µ
n
be measures of , n ∞. Suppose that
"
i
#, i = 1,…,n and that "
i
0 for all i. Show that
$
"
%
&
%
'
%()
(A): ="
%
&
%
(A).
5. Let Ω be a set and be the power set of Ω. Let x* and define +
,
on by +
,
(A):
=1
A
(x) for A . Show that +
-
is a measure on .
6. Determine which of the following collections of subsets of # is a semi-ring
i) .
)
={(ab): -∞/01}
ii) .
2
={(ab]: -∞/01}
Hint: . is called a semi-ring if (i) 3., (ii) A45. for A, B. and
(iii) A, B., then A65 is a finite disjoint union of elements of ..
7. Show that .
)
and .
2
in question 6 (.
)
) = (.
2
)
8. Let 7 be a ring of subsets of a set X and A :=78{(X4A
0
):A7}. Show that A is
an algebra.
www.schoolfaqs.net
9. Let (Ω,, µ) be a measure space and 7: ={A : µ(A)}. Show that 7 is a
ring.
10. Let (Ω,, µ) be a measure space and (*
9
,
:
&;) be the completion of (Ω,, µ), then
for a set of E.E
:
if and only if there exist sets A, C such that
A E C and µ(C6A) =0
11. Let Ω be a set which is not countable. Show that
(i) 7:={A *:is finite}is a ring
(ii) (7) = {A <Ω:A is countable or A
c
is countable}
(iii) µ
0
defined on 7 by µ
0
(B) = 0 for all B<7 is a pre-measure on 7
(iv) for all r 0 and µ
r
defined on (R) by µ
r
(A):=
!=>!/1?
=>!/1?
, µ
r
67 = µ
0
(v) µ
0
is not -finite on 7
12. Let Ω be a set which is not countable and 7 :={A *: A is finite or A
C
is finite}.
Let µ
1
and µ
2
be defined on 7 by µ
1
(A) =
!
µ
2
(A) =
!
@
i) Show that µ
1
and µ
2
are pre-measured on 7
ii) Determine >
)
A
and >
2
A
as well as
B
C
A
and
B
D
A
iii) Say whether or not, (and with justification) µ
i
has a unique extension to
(R) and
B
C
A
13. Let (Ω) be a measurable space and let and v be measurable on such that µ(Ω)
= v(Ω) . Show that {A: v(A)= µ(A)} is a Dynkin system
14. Which of the following are outer measures on E(#)?
(i) >
)
A
(A) :=
1>!FF
(ii) >
2
A
(A) :=
<
(iii)>
)
A
(A) :=
G
H3
3/!F1>!FF
In each case determine
B
C
A
if >
)
A
is an outer measure
15. Let {µ
n
} be a sequence of measures on a -algebra of subsets of a set Ω such
that µ
n
(A) 0 µ
n + 1
(A) for all A. Define µ on by µ(A):=sup{µ
n
(A):nI}.
Show that µ is a pre-measure on
16. Let Ω be a set and A be an algebra of subsets of Ω. Show that is µ and v are
measures on (A) such that (A) 0 v(A) for all A A, then µ(A)0 v(A) for all A (A)
17. Let Ω = I and := E(Ω). Define µ: J[-∞∞], AK
LH
M
$
)
2
N
'O
!
@
is µa measure?
www.schoolfaqs.net
