Measure Theory: Master the Fundamentals of Measure and Integration

Question 1

Let (X, Σ) be a measurable space and let μ be a measure on Σ. Which of the following sets is not necessarily measurable?

Accepted Answer

The set of all uncountable subsets of X

Answer

The set of all rational numbers in X

Answer

The set of all irrational numbers in X

Answer

The set of all closed subsets of X

Question 2

Let (X, Σ, μ) be a measure space. Which of the following is not a property of the integral of a non-negative function f?

Accepted Answer

It is always finite

Answer

It is always non-negative

Answer

It is always equal to the measure of the set {x ∈ X : f(x) > 0}

Answer

It is always less than or equal to the measure of the set {x ∈ X : f(x) > 0}

Question 3

Let (X, Σ, μ) be a measure space. Which of the following is a sufficient condition for f ∈ L¹(X, Σ, μ)?

Accepted Answer

f is measurable and |f| is integrable

Answer

f is measurable and f is bounded below

Answer

f is continuous and f is bounded above

Answer

f is differentiable and f is bounded

Question 4

Let (X, Σ, μ) be a measure space. Which of the following is true about the product measure μ × λ?

Accepted Answer

It is always a complete measure

Answer

It is always a σ-finite measure

Answer

It is always a regular measure

Answer

It is always a unique measure

Question 5

Identify the essential property of conditional expectation:

Accepted Answer

Linearity

Answer

Non-negativity

Answer

Monotonicity

Question 6

Let μ be a measure on a σ-algebra of subsets of X. What is the definition of the support of μ?

Accepted Answer

The smallest set A such that μ(X ∖ A) = 0

Answer

The largest set A such that μ(A) > 0

Answer

It is unique

Answer

It is always a closed set

Question 7

Which of the following is not a σ-algebra?

Accepted Answer

The collection of all subsets of the real line with finite measure

Answer

The power set of a set

Answer

The collection of all open intervals in the real line

Answer

The collection of all Borel sets in the real line

Question 8

Which of the following properties does NOT hold for a product measure?

Accepted Answer

It is equivalent to the sum of the individual measures

Answer

It is the unique translation-invariant measure on the product space

Answer

It is countably additive

Answer

It is complete

Question 9

Which of the following is NOT a convergence theorem specifically used in measure theory?

Accepted Answer

Chain rule for differentiation

Answer

Monotone convergence theorem

Answer

Fatou's lemma

Answer

Dominated convergence theorem

Question 10

In a measure space (X, Σ, μ), which set is not guaranteed to be measurable?

Accepted Answer

A set with finite measure

Answer

X

Answer

The empty set

Answer

A union of countably many measurable sets

Question 11

The integral of a non-negative function f over a measure space (X, Σ, μ) is defined as:

Accepted Answer

∫ f dμ = sup{∫ s dμ : s ≤ f, s simple}

Answer

∫ f dμ = inf{∫ u dμ : u ≥ f, u simple}

Answer

∫ f dμ = ∫ s dμ, where s is a simple function such that s ≤ f

Answer

∫ f dμ = ∫ s dμ, where s is a simple function such that s ≥ f

Question 12

A measure space in which every subset of X is measurable is known as:

Accepted Answer

Complete measure space

Answer

Discrete measure space

Answer

Perfect measure space

Answer

Trivial measure space

Question 13

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

Accepted Answer

µ(A) = 0 whenever A ⊆ B and µ(B) = 0 (sigma-additivity)

Answer

µ(∅) = 0

Answer

µ(E ∪ F) ≤ µ(E) + µ(F) for all E, F ∈ M (additivity)

Answer

µ(E) ≥ 0 for all E ∈ M (non-negativity)

Question 14

Let f: X → Y be a measurable function between two measure spaces (X, M, µ) and (Y, N, ν). Which statement concerning the preimage of a measurable set is true?

Accepted Answer

The preimage of any measurable set in Y is a measurable set in X.

Answer

The preimage of any measurable set in X is a measurable set in Y.

Answer

The image of any measurable set in Y is a measurable set in X.

Answer

The image of any measurable set in X is a measurable set in Y.

Question 15

Which property of the integral of a non-negative measurable function is incorrect?

Accepted Answer

If ∫ f dµ = ∞, then f(x) = ∞ for almost every x in X.

Answer

For any non-negative measurable function f, ∫ f dµ ≥ 0.

Answer

For any non-negative measurable function f and any constant c ≥ 0, ∫ c f dµ = c ∫ f dµ.

Answer

For any two non-negative measurable functions f and g, ∫ (f + g) dµ = ∫ f dµ + ∫ g dµ.

Question 16

What is the definition of the support of a measurable function?

Accepted Answer

The set of all points where the function takes non-zero values.

Answer

The set of all points where the function takes zero values.

Answer

The set of all points where the function is infinite.

Answer

The set of all points where the function is finite.

Question 17

Which of the following is NOT a necessary condition for the existence of a derivative of a measurable function?

Accepted Answer

The function is integrable.

Answer

The function is continuous almost everywhere.

Answer

The function is absolutely continuous.

Answer

The function is differentiable almost everywhere.

Question 18

Which property is NOT satisfied by a measure?

Accepted Answer

Non-countability

Answer

Non-negativity

Answer

Transitivity

Answer

Additivity

Question 19

Let f: X → Y be a measurable function between two measure spaces (X, Σ, μ) and (Y, Γ, ν). Which of the following functions is also measurable?

Accepted Answer

f ∘ g

Answer

f ∪ g

Answer

f ∩ g

Question 20

Let (X, Σ, μ) be a measure space and let f : X → [0, ∞] be a measurable function. Which of the following is a necessary and sufficient condition for f to be integrable?

Accepted Answer

f ∈ L¹(X, Σ, μ)

Answer

f ∈ L²(X, Σ, μ)

Answer

f ∈ L∞(X, Σ, μ)

Answer

f is continuous on X

Question 21

Let (X, Σ, μ) be a measure space and let f: X → R be an integrable function. Which of the following is equal to ∫Xf(x)dμ?

Accepted Answer

The expected value of f with respect to μ

Answer

The variance of f with respect to μ

Answer

The median of f with respect to μ

Answer

The mode of f with respect to μ

Question 22

Let (X, Σ, μ) be a measure space and let f : X → R be an integrable function. Which of the following is true?

Accepted Answer

∫Xf(x)dμ = ∫X+f(x)dμ + ∫X-f(x)dμ

Answer

∫Xf(x)dμ = ∫Xf'(x)dμ

Answer

∫Xf(x)dμ = ∫Xf²(x)dμ

Answer

∫Xf(x)dμ = ∫X|f(x)|dμ