Integration Theory-Ii - BU 2024 Question Paper

Download Original PDF

Get the official Barkatullah University print version scanned document.

🤝 Help Your Juniors!

Have previous year question papers that aren't on our website? Help the next batch of students by sending them to us! With your consent, we will proudly feature your name as a Top Contributor on our platform.

Submit Papers 📩

OP-511

M. Sc. (REG/PVT/ATKT) Examination, 2024

(Fourth Semester)

MATHEMATICS

Paper XII

Integration Theory-II

Time : 3 Hours]

[Maximum Marks :

Reg. 85

Pvt. 100

Note : Attempt all Sections as directed.

Section A

(Objective Type Questions)

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 ?

(a) f&sup+; = max [-f, 0]

(b) f = f&sup+; - f&sup-;

(d) f&sup+; = max {f, 0}

(ii)

Let μ and ν be measures on a measurable space (x, m). Which of the following conditions implies that μ is absolutely continuous with respect to ν ?

(a) For every set E such that ν(E) > 0, μ(E) > 0

(b) For every set E such that μ(E) > 0, ν(E) > 0

(d) For every set E such that μ(E) = 0, ν(E) = 0

(iii)

Fubini's theorem is concerned with which aspect of product measures ?

(a) Differentiation

(b) Integration

(d) Borel sets.

(iv)

Which of the following statements about the Borel measure on R is true ?

(a) The Borel measure is always finite.

(b) The Borel measure is Translation variant.

(d) The Borel measures assigns zero measure to every set.

(v)

Let X be a locally compact Hausdorff space and μ₁, μ₂ be Radon measures on B(X) for which

∫ f dμ₁ = ∫ f dμ₂ for all f ε C&supc;(x).

Then which of the following is true ?

(a) μ₁ > μ₂

(b) μ₁ < μ₂

(d) None of the above

Section B

(Short Answer Type Questions)

5×5=25

Define Continuity of Integration.

Define Positive linear functional on C&supc;(x).

Define absolute continuity of measure with example.

Define Signed measure on a measurable space with example.

If (ε, A₁, H) 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) ν (B) for all A ε A₁ and B ε A₂.

Write the statement of the Tonelli theorem.

Define Borel measure on R.

Define regularity of Lebesgue measure on R&supn;.

Write statement of the Riesz-Markov theorem.

10.

For a subset E of R&supn;, there is a G_δ subset G of R&supn; such that E⊂G and

μ*(G/E) = 0

Using Fubini's theorem, verify

∫₀¹∫₀¹ (x² - y²)/(x²+y²)² dx dy ≠ ∫₀¹∫₀¹ (x² - y²)/(x²+y²)² dy dx

Let μ be a Borel measure on B(I). Then its cumulative distribution function g_μ is increasing and continuous on the right. Conversely, each function g : I → R that is increasing and continuous on the right is the cumulative distribution function of a unique Borel measure μ_g on B(I).

Section C

(Long Answer Type Questions)

5×10=50

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 mλ < δ and ∫_A f < ε.

State and prove Lebesgue Dominated convergence theorem.

State and prove Vitali convergence theorem.

State and prove Lebesgue Decomposition theorem.

Prove Theorem of Fubini.

11.

Show that sum of two Radon measure is also a Radon measure.

Prove the theorem of Riesz-Markov.