LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – NOVEMBER 2003

ST-1801/S716 – MEASURE THEORY

06.11.2003 Max:100 marks

1.00 – 4.00

SECTION-A

Answer ALL the questions. (10×2=20 marks)

For a sequence {A_n} of sets, if A_n A, show that .
Define a monotone increasing sequence of sets and give its limt.
Show that a s – field is a monotone class.
Define the indicator function of a set A.
Show that the set rational numbers is a Borel set.
If X is a simple function, show that is a simple function.
If X₁ and X₂ are measurable functions with respect to prove that max { X_1,X₂} is measurable w.r.t .
If = {1,2,3,4}, is the power set of , μ {f} = 0, μ {1} = , μ {1,2} = ,

μ {1,2,3} = μ (W) = 1, is μ a measure on (W, )?

If μ is a measure, show that μ ≤ .
If = [0,1] and μ is the Lebesgue measure, write down the value of , where C is the set of rationals, A = [0, 3/4] and B = [1/2, 1].

SECTION-B

Answer any FIVE questions. (5×8=40 marks)

Prove that there exists a unique and minimal s-field on a given non – empty class of sets.
Define Borel s – field of subsets of real line. Show that the minimal s – field generated by the class of all open intervals is a Borel s – field. (2+6)
a) Define a finitely additive and a countably additive set functions.
b) Let W = {-3, -1, 0, 1, 3} and for A W, let l (A) = with l¹ = min (l, O), show that l is not even finitely additive.
If l is an extended real valued s – additive set function on a ring Â such that l(A) > – for every A Î Â, show that l is continuous at every set A Î Â.
If X₁ and X₂ are measurable functions w.r.t show that (X₁ + X₂) is also measurable w.r.t. prove that lim inf X_nis measurable w.r.t .
Define the Lebesgue – Stieltjes (LS) measure induced by a distribution function F on IR. If μ is the LS measure induced by

F(x) = 1 – e^-x if x > 0

if x ≤ 0,

then find (a) μ (0, 2) (b) μ [-1, +1] and (c) μ (A), where A = {0, 1, 2, 3, 4}. (2+6)

Show that a measure on a s – field can be extended to a complete measure.
State and establish Fatou’s lemma.

SECTION-C

Answer any TWO questions. (2×20=40 marks)

a) Distinguish between (i) a ring and a field (ii) a ring and a s – ring.
b) Define the minimal s-field containing a given class of sets. Give an example.
c) Show that the inverse image of a s-field is a s-field.
a) Define (i) extension of a measure (ii) completion of a measure. (6)
b) State and prove the Caratheodory extension theorem. (14)
a) Prove that if 0 ≤ X_n X, then . (8)
b) If X and Y are measurable functions on a measure space, show that

. (12)

a) If X ≥ 0 is an integrable function, prove that j (A) = A a measurable set,

defines a measure, which is absolutely continuous with respect to the measure m. (10)

b) State and prove the Lebesgue “dominated” convergence theorem. Is the

“denominated” condition necessary? Justify your answer. (10)

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – STATISTICS

FIRST SEMESTER – NOVEMBER 2004

ST 1902 – MEASURE THEORY

03.11.2004 Max:100 marks

9.00 – 12.00 Noon

SECTION – A

Answer ALL the questions (10 ´ 2 = 20 marks)

Let {A_n, n ≥ 1} be a sequence of subsets of a set W. Show that lim inf A_n C lim sup A_n.
Define minimal s – field.
What is a set function.?
Give an example of a counting measure.
Show that any interval is a Borel set but Borel set need not be an interval.
Define an Outer measure.
Define Lebesgue – Stieltjes measure.
Show that a composition of measurable functions is measurable.
Define a simple function with an example.
State Borel-Cantelli lemma.

SECTION – B

Answer any FIVE questions. (5 ´ 8 = 40 marks)

If {A_i, i ≥ 1} is a sequence of subsets of a set W then show that

(A_i ).

If D is a class of subsets of W and A C W, we denote D A the class {B A½B Î D}. If the minimal s – field over D is _W (D), Show that _A (D A) = _W (D)

Let ₀be a field of subsets of W. Let P be a probability measure on ₀. Let {A_n, n ≥ 1} and {B_n, n ≥ 1} be two increasing sequences of sets such that . Then show that

State and establish monotone class theorem.

If h and g are IB – measurable functions, then show that max {f, g} and min {f, g} are also IB – measurable functions.

If m is a measure on (W, ) and A₁, A₂,… is a sequence of sets in , Use Fatou’s lemma to show that
m
If m is finite, then show that m .

Define absolute continuity of measures. Show that l < < m if and only if < < m.

State Radon – Nikodym theorem. Mention any two applications of this theorem to probability / statistics.

SECTION – C

Answer any TWO questions (2 ´ 20 = 40 marks)

a) Let {x_n} be a sequence of real numbers, and let A_n = (-¥, x_n). What is the connection

between sup x_n and Similarly what is the connection between

inf x_n and inf A_n.

Show that every s – field is a field. Is the converse true? (8+12)

a) Let W be countably infinite set and let consist of all subsets of W. Define

0 if A is finite

m (A) = ¥ if A is infinite.

Show that m is finitely additive but not countably additive.
Show that W is the limit of an increasing sequence of sets A_n with

m (A_n) = 0 “_n but m (W) = .

b) Show that a s – field _s is a monotone class but the converse is not true. (7+7+6)

a) State and establish Caratheodory extension theorem.

b) If exists and C Î IR then show that = . (12+8)

a) State and establish extended monotone convergence theorem.

b) State and establish Jordan – Hahn Decomposition theorem. (10+10)