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M.A./M.Sc. IIIrd Semester (Reg./Pvt./ATKT)
Examination, 2021-22
Maths
Paper - XII
Integration Theory-I
Reg.= 85
Pvt.= 100
Note :- Attempt all questions.
SECTION - 'A'
Objective Type Questions
5×3=15
1. Choose the correct answer :
(i)
Let (X, M) be a measurable space, If μ and ν are measures defined on M, then the set function λ defined on M by ............. also is a measure.
(a) λ(E) = μ(E) + ν(E)
(b) λ(E) = μ(E) - ν(E)
(c) λ(E) = μ(E) × ν(E)
(d) None of these
(ii)
A decomposition of X into the union of two disjoint sets A and B for which A is positive for ν and B is negative is called a Hahn decomposition for ............
(a) X
(b) ν
(c) A
(d) None of these
(iii)
Let S be a semiring of subsets of a set X. Define S' to be the collection of unions of finite disjoint collections of sets in S. Then S' is .......... with respect to the formation of relative complements.
(a) Open
(b) Bounded
(c) Closed
(d) None of these
(iv)
The sum of two extended real-valued functions is not defined at points where the functions take infinite value of ............ sign.
(a) Same
(b) Positive
(c) Negative
(d) Opposite
(v)
Let f and g be nonnegative measurable functions on X for which g ≤ f a.e. on X, then f = g a.e. on X if and only if ............
(a) ∫X g dμ = ∫X f dμ
(b) ∫X g dμ > ∫X f dμ
(c) ∫X g dμ < ∫X f dμ
(d) None of these
SECTION - 'B'
Short Answer Type Questions
5×5=25
1. Define measurable sets, measurable space and continuity of measure with example.
OR
Let (X, M) be a measurable space. If μ and ν are measures on M and μ ≥ ν then there is a measure λ on M for which μ = ν + λ
2. Let ν be a signed measure on the measurable space (X, M). Then every measurable subset of a positive set is itself positive and the union of a countable collection of positive sets is positive.
OR
State and prove Hahn's Lemma.
3. Define outer measure and Caratheodory measure with example.
OR
Show that any measure that is induced by an outer measure is complete.
OR
State and prove Caratheodory - Hahn Theorem.
4. Let E be a measurable subset of X and f an extended real-valued function on X. Show that f is measurable if and only if its restrictions to E and X ~ E are measurable.
OR
State and prove simple Approximation Theorem.
OR
Show that a set function on a σ-algebra is a measure if and only if it is a premeasure.
5. State and prove Chebychev's Inequality.
OR
State and prove Fatou's Lemma.
SECTION - 'C'
Long Answer Type Questions
9×5=45
1. State and prove Borel - Cantelli Lemma.
OR
Let (X, M) be a measurable space. If μ and ν are measures defined on M, then the set function λ defined on M by λ(E) = μ(E) + ν(E) also is a measure.
2. State and prove Hahn Decomposition Theorem.
OR
State and prove Jordan Decomposition Theorem.