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Total No. of Questions : 11
Total No. of Printed Pages : 07
State and prove Integral Comparison test.
OP-511
3. Define absolute continuity of measure with example.
M. Sc. (REG/PVT/JATKT) Examination, 2024
Or
(Fourth Semester)
Define Signed measure on a measurable space with example.
MATHEMATICS
4. If (ε, A₁, R) and (y, A₂, K) are measure spaces, then there exists a measure π defined as A = A₁ × A₂. Such that π (A × B) = μ (A) v (B) for all A ∈ A₁ and B ∈ A₂.
Paper XII
Or
Integration Theory-II
Write the statement of the Tonelli theorem.
Time : 3 Hours]
[Maximum Marks :
Reg. 85
Pvt. 100
5. Define Borel measure on R.
Note : Attempt all Sections as directed.
Or
Define regularity of Lebesgue measure on Rⁿ.
Section A
(Objective Type Questions)
6. Write statement of the Riesz-Markov theorem.
5×2=10
1.
Choose the correct answer :
(i) If (x, m) be a measurable space and f a measurable function on x, then which of the following is not correct ?
(ii) Let μ and v be measures on a measurable space (x, m). Which of the following conditions implies that μ is absolutely continuous with respect to v ?
(iii) Fubini's theorem is concerned with which aspect of product measures ?
(iv) Which of the following statements about the Borel measure on R is true ?
(v) Let X be a locally compact Hausdorff space and μ₁, μ₂ be Radon measures on B(x) for which ∫_x f dμ₁ = ∫_x f dμ₂ for all f ∈ C₀(x). Then which of the following is true ?
Or
Define Positive linear functional on Cₖ(x).
Section C
5×10=50
7.
Let f be a non-negative function which is integrable over a set E. Prove that given ε > 0 there is a δ > 0 such that for every A ⊂ E with m(A) < δ and
∫A f dμ < ε.
Or
State and prove Lebesgue Dominated convergence theorem
8.
State and prove Vitali convergence theorem.
Or
State and prove Lebesgue Decomposition theorem.
9.
Prove Theorem of Fubini.
Section B
5×5=25
(Short Answer Type Questions)
2.
Define Continuity of Integration.
10.
Using Fubini's theorem, verify
For a subset E of Rⁿ, there is a G_δ, subset G of Rⁿ such that E ⊂ G and μ*(G/E)=0
Or
Let μ be a Borel measure on B(1). Then its cumulative distribution function g_μ is increasing and continuous on the right. Conversely, each function g : 1 → R that is increasing and continuous on the right is the cumulative distribution function of a unique Borel measure μ_g on B(1).
11.
Show that sum of two Radon measure is also a Radon measure.
Or
Prove the theorem of Riesz-Markov.