Unit 7 - Lebesgue measure and invariance properties

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10/07/2020 Measure Theory - - Unit 7 - Lebesgue measure and invariance properties

https://onlinecourses.nptel.ac.in/noc20_ma02/unit?unit=48&assessment=102 1/4

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» Measure Theory (course)

Unit 7 - Lebesgue measure and invariance properties

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Course outline

How does an NPTEL online course work?

Sigma algebras, Measures and Integration

Integration and convergence theorems Outer measure

Lebesgue measure and its properties

Lebesgue measure and positive Borel measures on locally compact spaces

Lebesgue measure and invariance properties

Due on 2020-03-11, 23:59 IST.

1 point 1)

1 point 2)

Week 6 Assessment

The due date for submitting this assignment has passed.

As per our records you have not submitted this assignment.

Which of the following sets are of Lebesgue measure zero?

Any countable subset of Cantor set

The set of irrationals in the Cantor set

dimensional subspace of where No, the answer is incorrect.

Score: 0

Accepted Answers:

Any countable subset of Cantor set

The set of irrationals in the Cantor set dimensional subspace of where

Let be an invertible linear map. Which of the following are correct?

If is a Lebesgue set then is a Lebesgue set

If is a Borel set then is a Borel set No, the answer is incorrect.

Score: 0

Accepted Answers:

ℝ

𝑘− ℝ

^𝑛

, 𝑘 < 𝑛

ℝ

𝑘− ℝ

^𝑛

, 𝑘 < 𝑛

𝑇 : ℝ

^𝑛

→ ℝ

^𝑛

𝐸 𝑇(𝐸)

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L^p spaces and completeness

Product spaces and Fubini's theorem

Applications of Fubini's theorem and complex measures

Complex measures and Radon-Nikodym theorem

Radon-Nikodym theorem and applications

Riesz

representation theorem and Lebesgue

Lebesgue measure via Riesz representation theorem (unit?

unit=48&lesson=49) Construction of Lebesgue measure (unit?

unit=48&lesson=50) Invariance

properties of Lebesgue measure (unit?

unit=48&lesson=51) Linear

transformations and Lebesgue measure (unit?

unit=48&lesson=52) Cantor set (unit?

unit=48&lesson=53) Cantor function (unit?

unit=48&lesson=54) Quiz : Week 6 Assessment (assessment?

name=102)

1 point 3)

1 point 4)

1 point 5)

1 point 6)

If is a Lebesgue set then is a Lebesgue set If is a Borel set then is a Borel set

Which of the following are true?

The outer measure on is translation invariant

The Lebesgue measure on is translation invariant

If is any Borel measure on and for all compact set then is a constant multiple of the Lebesgue measure

No, the answer is incorrect.

Score: 0

Accepted Answers:

The outer measure on is translation invariant The Lebesgue measure on is translation invariant

Let be a Borel measure on such that for all Borel sets and Then which of the following are always correct?

is the zero measure

is a constant multiple of the Lebesgue measure

Counting measure on satisfies the property for all and No, the answer is incorrect.

Score: 0

Accepted Answers:

Counting measure on satisfies the property for all and Let be a linear transformation. Which of the following are correct?

If is singular, then is a Lebesgue set for all

If is invertible, then is a Lebesgue set for all

If is invertible, then is a Lebesgue set for all Lebesgue sets No, the answer is incorrect.

Score: 0

Accepted Answers:

If is singular, then is a Lebesgue set for all

If is invertible, then is a Lebesgue set for all Lebesgue sets

Let be a positive measurable function and let denote the Lebesgue measure on Which of the following are true?

for all and

for all

for all linear maps

N h i i

𝐸 𝑇(𝐸)

𝑚

∗

ℝ

𝑚 ℝ

𝜇 ℝ 𝜇(𝐾) < ∞ 𝐾, 𝜇

𝑚

∗

ℝ

𝑚 ℝ

𝜇 ℝ 𝜇(𝐴 + 𝑛) = 𝜇(𝐴) 𝐴

𝑛 ∈ ℤ.

𝜇 𝜇

ℝ 𝜇(𝐴 + 𝑛) = 𝜇(𝐴) 𝐴 𝑛 ∈ ℤ.

𝐴 : ℝ

^𝑛

→ ℝ

^𝑛

𝐴 𝐴(𝐸) 𝐸 ⊂ ℝ

^𝑛

𝑓 : ℝ

^𝑛

→ ℝ 𝑚

ℝ.

𝑓 𝑑𝑚 = 𝑓 𝑑𝑚

∫

_𝐸+𝑥₀

∫

_𝐸

𝐸 ∈ ( ) 𝑅

^𝑛

𝑥

0

∈ ℝ

^𝑛

𝑓(𝑥 + ) 𝑑𝑥 = 𝑓 𝑑𝑚

∫

_ℝ^𝑛

𝑥

0

∫

_ℝ^𝑛

𝑥

0

∈ ℝ

^𝑛

𝑓(𝐴𝑥) 𝑑𝑥 = 𝑓(𝑥) 𝑑𝑥

∫

_ℝ𝑛

∫

_ℝ𝑛

𝐴 : ℝ

^𝑛

→ ℝ

^𝑛

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differentiation theorem

Weekly Feedback forms

Video download

1 point 7)

1 point 8)

1 point 9)

1 point 10)

Score: 0

Accepted Answers:

for all Which of the following are true?

If is a dense open set then

If is open then

If is closed and has no interior then No, the answer is incorrect.

Score: 0

Accepted Answers:

If is open then

Which of the following sets are of Lebesgue measure zero?

where is the Cantor set

Countable union of lines passing through the origin in No, the answer is incorrect.

Score: 0

Accepted Answers:

where is the Cantor set

Countable union of lines passing through the origin in

Which of the following sets have positive Lebesgue measure?

Any unbounded set in

Any unbounded closed set in

Score: 0

Accepted Answers:

Which of the following are correct?

The Cantor set is a closed set The Cantor set is a perfect set

The complement of Cantor set in has positive Lebesgue measure The Cantor set is uncountable

Score: 0

Accepted Answers:

𝑓(𝑥 + ) 𝑑𝑥 = 𝑓 𝑑𝑚

∫

_ℝ^𝑛

𝑥

0

∫

_ℝ^𝑛

𝑥

0

∈ ℝ

^𝑛

𝑂 ⊂ [0, 1] 𝑚(𝑂) = 1

𝑂 ⊂ [0, 1] 𝑚(𝑂) > 0

𝐹 ⊂ [0, 1] 𝑚(𝐹) = 0

𝑂 ⊂ [0, 1] 𝑚(𝑂) > 0

𝐶 × ℝ ⊂ ℝ

²

𝐶 ⊂ [0, 1]

ℚ × ℝ ⊂ ℝ

²

ℝ

²

𝐶 × ℝ ⊂ ℝ

²

𝐶 ⊂ [0, 1]

ℚ × ℝ ⊂ ℝ

²

ℝ

²

ℝ

²

ℝ

²

(ℝ ∖ ℚ) × ℝ ⊂ ℝ

²

(ℝ ∖ ℚ) × ℝ ⊂ ℝ

²

[0, 1]

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The Cantor set is a closed set The Cantor set is a perfect set

The complement of Cantor set in has positive Lebesgue measure The Cantor set is uncountable

[0, 1]