Lebesgue Measure And Integration-Ii - BU 2024 Question Paper

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OP-484

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

(Second Semester)

MATHEMATICS

Paper-II

Lebesgue Measure and Integration-II

Time: 3 Hours] [Maximum Marks :

Reg. : 85 Pvt. : 100

Note : Attempt the questions of all Sections as directed.

Section A

(Objective Type Questions)

5×5=10

Note : Attempt all questions. All questions carry equal marks.

Choose the correct answer :

(i) Every countable set has measure :

(a) Countable

(b) φ

(d) 1

(ii) If f⁺ and f^- are positive and negative parts of any real function f, then f is measurable if :

(a) f⁺ is measurable

(b) f^- is measurable

(d) Both (a) and (b)

(iii) If f on [0, 1] is defined by :

then :

(a) f ∈ BV[0,1]

(b) f ∉ BV[0,1]

(d) None of the above

(iv) Every convex function on an open interval is :

(a) Continuous

(b) Discontinuous

(d) None of the above

(v) If f_n → f is measure and g_n → g is measure, then :

(a) f_n + g_n → f + g is measure

(b) αf_n → αf is measure for α ∈ R

(d) All of the above

Section B

(Short Answer Type Questions)

5×5=25

Note : Attempt all questions. All questions carry equal marks.

Define Lebesgue out measure m*(A) for a set A and prove that for A ⊂ B, m*(A) ≤ m*(B).

If f is a measurable function, then show that {x : f(x) = α} is measurable for each extended real number α.

If f is a non-negative measurable function, then prove that ∫₀^a f = 0 a.e. if and only if ∫₀^a f dx = 0.

If f is integrable, then show that:

| ∫ f dx | ≤ ∫ |f| dx.

Define four derivatives at x of an extended real valued function f, which is finite at x and defined in an open interval containing x.

Show that BV[a, b] is a vector space over R.

Let f, g ∈ L_p(μ) and let a, b be constants then prove that a f + b g ∈ L_p(μ).

Let p ≥ 1, and let f, g ∈ L_p(μ), then prove that :

(∫ |f + g|^p dμ)^1/p ≤ (∫ (1+1)^p dμ)^1/p (∫ |g|^p dμ)^1/p.

If a sequence of measurable functions converges in measure, then prove that the limit function is unique a.e.

If f_n → f a.u., then prove that f_n → f is measure.

Section C

(Long Answer Type Questions)

5×10=50

Note : Attempt all questions. All questions carry equal marks.

Prove that the class M of Lebesgue measurable sets is a σ-algebra.

State and prove Fatou's Lemma.

If f is Riemann integrable and bounded over the finite interval [a, b], then prove that f is integrable and

∫_a^b R f dx = ∫_a^b f dx.

If f ∈ BV [a, b] where a and b are finite, then prove that :

(i) f is differentiable a.e.

(ii) the derivative is finite a.e.

If f is finite valued monotone increasing function defined on the finite interval [a, b], then prove that

f' is measurable and ∫_a^b f' dx ≤ f(b) - f(a).

10.

State and prove Jensen's Inequality.

If 1 ≤ p < ∞ and (f_n) is a sequence in L^p (μ) such that :

||f_n - f_m||_p → 0 as n, m → ∞,

then prove that there exists a function f and a subsequence (n_k) such that lim f_{n_k} = f a.e.

Also prove that f ∈ L^p (μ) and lim ||f_n - f|| = 0.

11.

If (f_n) is a sequence of measurable functions which is fundamental in measure, then prove that there exists a measurable function f such that f_n → f is measure.

Let f_n → f a.e. If μ(X) < ∞ or for each n ∫ |f_n| ≤ g, an integrable function then prove that :

f_n → f a.u.