Lebesgue Measure Integration-Ii - BU 2022 Question Paper

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Total No of Questions 11

[Total No. of Printed Pages : 6

GH-484

M.A./M.Sc. IInd Semester (Reg./Pvt./ATKT)

Examination, 2022

Maths

Paper - II

Lebesgue Measure & Integration-II

Time : 3 Hours]

[Maximum Marks :
Reg.= 85
Pvt.= 100

Note- Attempt all the questions.

SECTION - 'A'

Objective Type Questions

Choose the correct option :

Singleton set {x} has its measure
1. 0
2. 1
3. 1/2
4. None of these
If f is Lebesgue integrable on E and A ⊂ E is measurable then.
1. ∫_A f = ∫_E f
2. ∫_A f ≤ ∫_E f
3. ∫_A f ≥ ∫_E f
4. None of these
Let f(x) = |x| then at x = 0
1. D⁺ = 0 = D^-
2. D₊ = 1, D_- = 0
3. D⁺ = 1, D^- = -1
4. None of these
If f ∈ L^P (μ) then ||f||_P is given by
1. ( ∫|f|^P )
2. ( ∫|f|^P )^1/P
3. ( ∫|f|^P )^1/P
4. None of these
Suppose f_n → f in measure. Then for f_n² → f²
1. μ(x) < ∞
2. μ(x) = ∞
3. μ(x) = 0
4. None of these

SECTION - 'B'

Short Answer Type Questions

Define vebesgue outer measure with example.

Define measureable function with example.

Show that

∫₀¹ log 1_x dx = 9 ∑_n=1^∞ 1_(3n+1)²

Let f and g be non-negative measurable functions. If f ≤ g then prove that ∫fdx ≤ ∫gdx.

Define four derivatives for an extended real valued function f defined on an open interval.

If f ∈ L(a,b) then prove that F(x) = ∫_a^x f(t) dt in a continuous function on [a,b].

If f,g ∈ L¹(μ) and a, b are constant them prove that af + bg ∈ L¹(μ).

If f ∈ L¹(μ) and g ∈ L^∞ (μ) then prove that ||fg||₁ ≤ || f ||₁ || g ||_∞.

If f_n → f almost uniform then prove that f_n → f in measure.

SECTION - 'C'

Long Answer Type Questions

Prove that outer measure of an interval is its length.

Let E be a measureable set. Then for each y, prove that the set. E + y = {x + y : x ∈ E} is also measureable and m (E) = m (E + y).

Let {f_n: n=1, 2, .....} be a sequence of non - negative measurable functions. Then prove that

Lim inf ∫ f_ndx ≥ ∫ lim inf f_n dx

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

R ∫_a^b fdx = ∫_a^b fdx

Define positive, negative and total variation of a function f on [a, b]. If f is a function of bounded variation on [a, b] then prove that

f(b) - f(a) = P - N

and T = P + N

where P, N and T are positive, negative and total variatoin of f on [a, b].

10.

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

∫_a^b f'dx ≤ f (b) - f (a)

Prove that every convex function on an open interval is continuous.

11.

If {f_n} is a sequence of measurable functions which is fundamental in measure then prove that there exists a measureable function f such that f_n → f in 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.