Measure Theory: A Comprehensive Guide for BSc Mathematics Students

Question 1

Let (f: X ightarrow E) be a measurable function and (E in mathcal{M}) for a measure space ((X, Sigma, mu)). Prove that the set ({x in X : f(x) in E}) is measurable.

Accepted Answer

Since (f) is measurable, the set ({x in X : f(x) in E}) is also measurable.

Answer

If (f) is continuous, then the set ({x in X : f(x) in E}) is measurable.

Answer

If (E) is open, then the set ({x in X : f(x) in E}) is measurable.

Question 2

Which of the following is not a fundamental property of a measure space?

Accepted Answer

Every measurable function is continuous.

Answer

The measure of the empty set is zero.

Answer

The measure of a disjoint union of sets is equal to the sum of the measures of the sets.

Answer

The measure of a set is non-negative.

Question 3

Which of the following is a necessary condition for the integrability of a function?

Accepted Answer

The function is measurable.

Answer

The function is continuous.

Answer

The function is bounded.

Answer

The function is differentiable.

Question 4

Which of the following theorems relates integration and differentiation?

Accepted Answer

The Fundamental Theorem of Calculus.

Answer

The Chain Rule.

Answer

The Product Rule.

Answer

The Quotient Rule.

Question 5

Let (X, Σ, μ) be a measure space and let A, B ∈ Σ. Which of the following statements about the measure of A ∪ B is true?

Accepted Answer

If A ∩ B = ∅, then μ(A ∪ B) = μ(A) + μ(B).

Answer

If A ∩ B ≠ ∅, then μ(A ∪ B) = μ(A) + μ(B).

Answer

If A ⊂ B, then μ(A ∪ B) = μ(A) - μ(B).

Answer

If A ⊂ B, then μ(A ∪ B) = μ(B) - μ(A).

Question 6

When integrating a non-negative measurable function, what is the expected outcome?

Accepted Answer

A non-negative real number

Answer

A positive real number

Answer

A complex number

Answer

Infinity

Question 7

In the hierarchy of measures, which type is the most general?

Accepted Answer

Borel measure

Answer

Lebesgue measure

Answer

Counting measure

Answer

Hausdorff measure

Question 8

Which property is associated with sigma-finite measures?

Accepted Answer

Decomposable into a countable union of finite measures

Answer

Defined on an uncountable set

Answer

Absolutely continuous relative to Lebesgue measure

Answer

Singular relative to Lebesgue measure

Question 9

What is the primary function of the Radon-Nikodym Theorem?

Accepted Answer

Decomposing a measure into absolutely continuous and singular parts

Answer

Establishing the existence of a measure's derivative

Answer

Computing the integral of a function relative to a measure

Answer

Constructing a measure from a given density function

Question 10

Let f be a non-negative measurable function on a measure space (X, Σ, μ). Which of the following is equivalent to the statement that f is integrable?

Accepted Answer

∫ f dμ is finite.

Answer

∫ f dμ = 0.

Answer

f is bounded on X.

Answer

f is continuous on X.

Question 11

Let (X, Σ, μ) be a measure space. Which of the following is NOT a property of a measurable function f: X → R?

Accepted Answer

f is Riemann integrable.

Answer

f is measurable with respect to the Borel σ-algebra.

Answer

f is measurable with respect to the Σ-algebra.

Answer

The preimage of any open set in R is measurable in X.

Question 12

Let (X, Σ, μ) be a measure space. Which of the following statements is true about the integral of a non-negative measurable function f: X → R?

Accepted Answer

It may be infinite.

Answer

It is always finite.

Answer

It is always positive.

Answer

It is always zero.

Question 13

Let (X, Σ, μ) be a measure space. Which of the following statements is true about the derivative of a measurable function f: X → R?

Accepted Answer

It exists almost everywhere.

Answer

It exists everywhere.

Answer

It is always continuous.

Answer

It is always differentiable.

Question 14

Which of the following functions is NOT measurable?

Accepted Answer

The inverse sine function

Answer

The indicator function of a measurable set

Answer

The characteristic function of an interval

Answer

The floor function

Question 15

Let (X, Σ, μ) be a measure space and let f: X → R be a measurable function. The integral of f over X is defined as:

Accepted Answer

∫​X​f(x)dμ(x)

Answer

∫​R​dμ(f)

Answer

∫​X​f(x)μ(dx)

Answer

∫​dμ(f(x))

Question 16

Which of the following is a consequence of the Radon-Nikodym Theorem?

Accepted Answer

Every measure on a σ-finite measure space is absolutely continuous with respect to some other measure.

Answer

Every absolutely continuous measure is singular with respect to some other measure.

Answer

Every measure on a compact space is absolutely continuous with respect to Lebesgue measure.

Answer

Every singular measure is absolutely continuous with respect to some other measure.

Question 17

Let (X, Σ, μ) be a measure space. Which of the following is NOT a property of the outer measure defined by μ?

Accepted Answer

It is a measure.

Answer

It is monotone.

Answer

It is countably subadditive.

Answer

It is translation invariant.

Question 18

What is the relationship between the Lebesgue measure and the Borel measure on a Borel set?

Accepted Answer

They are equal.

Answer

The Lebesgue measure is always greater than the Borel measure.

Answer

The Borel measure is always greater than the Lebesgue measure.

Answer

They are not related.

Question 19

Which theorem guarantees that every function defined on a measure space can be approximated by a sequence of measurable functions?

Accepted Answer

Lusin's Theorem

Answer

Egoroff's Theorem

Answer

Stone-Weierstrass Theorem

Answer

Radon-Nikodym Theorem

Question 20

Let f be a measurable function on a measure space (X, Σ, μ). Which of the following statements is true?

Accepted Answer

The set of points where f is discontinuous has measure zero.

Answer

The set of points where f is continuous has measure one.

Question 21

Let f be a measurable function on a measure space (X, Σ, μ). What is true about the integral of f over X?

Accepted Answer

It is equal to the sum of the integrals of f over the disjoint subsets of X.

Answer

The integral of f over X is greater than or equal to the sum of the integrals of f over the disjoint subsets of X.

Question 22

Let f be a measurable function on a measure space (X, Σ, μ). Which statement about the set of points where f is equal to its essential supremum is true?

Accepted Answer

It has measure zero.

Answer

The set of points where f is not equal to its integral over X has measure zero.

Question 23

Which of the following is a property of a σ-finite measure space?

Accepted Answer

It can be represented as a countable union of finite measure spaces.

Answer

It is always complete.

Question 24

Which of the following is **not** a type of differentiation in measure theory?

Accepted Answer

Riemann differentiation

Answer

Lebesgue differentiation

Answer

Rademacher differentiation

Answer

Fréchet differentiation

Question 25

Let (X, Σ, μ) be a measure space. Which statement accurately describes the set of measurable functions on X?

Accepted Answer

It forms both a σ-algebra and a vector space.

Answer

It forms a σ-algebra.

Answer

It forms a vector space.

Answer

It forms neither a σ-algebra nor a vector space.

Question 26

Let f be a nonnegative function on a measure space (X, Σ, μ). Which of the following is a necessary and sufficient condition for f to be integrable?

Accepted Answer

The set {x ∈ X | f(x) = ∞} has μ-measure zero.

Answer

f is finite everywhere on X.

Answer

The integral of f over X is finite.

Answer

The range of f is bounded.

Question 27

Let (X, Σ, μ) be a measure space and let f and g be measurable functions on X. Which of the following statements is true?

Accepted Answer

The sum of two integrable functions is integrable.

Answer

The product of two integrable functions is integrable.

Answer

The quotient of two integrable functions is integrable.

Answer

The inverse of an integrable function is integrable.

Question 28

Let (X, Σ, μ) be a measure space. Which of the following sets is not measurable? (set of positive numbers, set of rational numbers, set where sine of x equals 1, set of natural numbers)

Accepted Answer

Set where sine of x equals 1

Answer

Set of positive numbers

Answer

Set of rational numbers

Answer

Set of natural numbers

Question 29

What is the integral of the function f(x) = x^2 over the interval [0, 1] with respect to the Lebesgue measure?

Accepted Answer

1/3

Answer

1/2

Answer

1

Answer

3/2

Question 30

Let (X, Σ, μ) be a measure space and let f be a measurable function. Which of the following is true? (Positivity of integrals is a fundamental property of integrals.)

Accepted Answer

If f is non-negative, then ∫_X f dμ ≥ 0

Answer

If f is non-negative, then ∫_X f dμ = 0

Answer

If f is non-negative, then ∫_X f dμ ≤ 0

Answer

If f is non-negative, then ∫_X f dμ = 1

Question 31

Let (X, Σ, μ) be a measure space and let f be a measurable function. Which of the following is true? (Properties of integrable functions are essential for understanding their behavior.)

Accepted Answer

If f is integrable, then ∫_X f dμ is finite

Answer

If f is integrable, then ∫_X f dμ = 0

Answer

If f is integrable, then ∫_X f dμ = ∞

Answer

If f is integrable, then ∫_X f dμ is a complex number

Question 32

Which of the following is not a property of a measure space?

Accepted Answer

Finite

Answer

Non-negative

Answer

Countably additive

Answer

Translation invariant

Question 33

Which of the following is a necessary condition for a function to be measurable?

Accepted Answer

Its range is a measurable set

Answer

It is continuous

Answer

It is bounded

Answer

It is differentiable

Question 34

Let (X, Σ, μ) be a measure space. Which of the following statements is true?

Accepted Answer

The set of all measurable sets in X is a σ-algebra

Answer

The set of all measurable sets in X is an algebra

Answer

The set of all measurable sets in X is a field

Answer

The set of all measurable sets in X is a lattice

Question 35

Which of the following is a sufficient condition for a function to be integrable?

Accepted Answer

It is bounded

Answer

It is continuous

Answer

Its range is a finite set

Answer

It is differentiable

Question 36

Which of the following is NOT a fundamental property of a measure space?

Accepted Answer

Uniqueness

Answer

Non-negativity

Answer

Countable additivity

Answer

Completeness

Question 37

In the definition of a measure space, which element is a mapping from a set to the real numbers?

Accepted Answer

The measure

Answer

The set of points

Answer

The set of measurable functions

Answer

The nonnegative function defined on a set

Question 38

What is the significance of the Lebesgue integral of the indicator function of a measurable set E?

Accepted Answer

It represents the measure of the set E

Answer

It represents the length of the set E

Answer

It represents the volume of the set E

Answer

It represents the cardinality of the set E

Question 39

Which of the following properties is not characteristic of the Lebesgue measure on the real line?

Accepted Answer

Continuity

Answer

Translation-invariance

Answer

Scaling-invariance

Answer

Countable additivity