Measure Theory Quizzes - Test Your Understanding

Question 1

Which of the following sets is not Lebesgue measurable?

Accepted Answer

The Vitali set

Answer

A bounded interval

Answer

A countable set

Answer

The Cantor set

Question 2

Differentiability of a function f at a point x does NOT require:

Accepted Answer

Local minimum or maximum at x

Answer

Continuity at x

Answer

Absolute continuity at x

Answer

Existence of f' at x

Question 3

The central concept in measure theory is:

Accepted Answer

Measurability

Answer

Integration

Answer

Differentiation

Answer

Topology

Question 4

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

Accepted Answer

∫f dμ

Answer

supremum of the set of integrals of measurable functions g such that g ≤ f

Answer

infimum of the set of integrals of measurable functions g such that g ≥ f

Answer

∫|f| dμ

Question 5

Given a set E in R and a Borel measure μ, which of the following statements about the Lebesgue measure of E is true?

Accepted Answer

It is equal to the sum of the lengths of the non-overlapping intervals in E.

Answer

It is equal to the area of the set E.

Answer

It is always finite.

Answer

It is always positive.

Question 6

What is the total variation of the measure μ defined on [0, 1] by μ([a, b]) = b - a?

Accepted Answer

1

Answer

0

Answer

2

Answer

∞

Question 7

What is the support of the measure μ defined on R by μ(E) = the number of rational numbers in E?

Accepted Answer

Q

Answer

R

Answer

The set of all subsets of R.

Answer

The empty set.

Question 8

How does measure theory contribute to the construction of probability spaces?

Accepted Answer

It allows for the precise definition of probability as a measure on a sample space.

Answer

It enables the direct calculation of event probabilities by summing set measures.

Answer

It restricts the concept of probability to infinite sample spaces.

Answer

It has no role in the construction of probability spaces.

Question 9

Let (X, Σ, μ) be a measure space and f a measurable function on X. What can we say about the set {x ∈ X: f(x) > c} for any real number c?

Accepted Answer

It is measurable.

Answer

It is measurable if and only if f is continuous.

Answer

It is measurable if and only if μ({x ∈ X: f(x) > c}) = 0.

Answer

It is not measurable if f is unbounded.

Question 10

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

Accepted Answer

(ℝ, ℬ(ℝ), λ), where ℝ is the set of real numbers, ℬ(ℝ) is the Borel σ-algebra on ℝ, and λ is the Lebesgue measure.

Answer

([0, 1], ℬ([0, 1]), δ0), where δ0 is the Dirac measure at 0

Question 11

Given a measure space (X, Σ, μ) and a measurable function f on X, what property does the integral ∫Xf dμ possess?

Accepted Answer

It is a linear operator.

Answer

It is a continuous operator.

Answer

It is a bounded operator.

Question 12

Which theorem is crucial for proving the existence of a Lebesgue measure on the real line?

Accepted Answer

Carathéodory's Extension Theorem

Answer

Riesz Representation Theorem

Answer

Radon-Nikodym Theorem

Answer

Fubini's Theorem

Question 13

Which of the following demonstrates a key application of measure theory in probability theory?

Accepted Answer

Defining the probability of events in a probability space.

Answer

Finding the area of a circle.

Answer

Solving differential equations.

Question 14

Which of the following is a property of Lebesgue measure on the real line?

Accepted Answer

σ-additivity

Answer

Translation invariance

Answer

Additivity for all subsets of the real line

Answer

Non-negativity

Question 15

Let f(x) be a non-negative function defined on a measurable set E. The Lebesgue integral of f over E with respect to Lebesgue measure is defined as the

Accepted Answer

supremum of the integrals of simple functions less than or equal to f

Answer

area under the graph of f over E

Answer

infimum of the integrals of simple functions greater than or equal to f

Question 16

Let X be a set, μ be a measure on X, and f be a real-valued function on X. A necessary and sufficient condition for f to be μ-integrable is that

Accepted Answer

f is measurable and the integral of the absolute value of f with respect to μ is finite

Answer

f is differentiable and μ(X) is σ-finite

Answer

f is continuous and μ(X) is finite

Answer

f is bounded and μ(X) is finite

Question 17

A key application of measure theory in probability is the

Accepted Answer

construction of probability spaces

Answer

approximation of the value of π

Answer

determination of the volume of geometric shapes

Answer

solution of differential equations

Question 18

Let μ be a σ-finite measure on a set X. Which of the following is true?

Accepted Answer

Not every measurable subset of X must have finite measure

Answer

There exists a measurable set E such that 0 < μ(E) < ∞

Answer

μ(X) must be infinite

Question 19

A consequence of the Dominated Convergence Theorem is that

Accepted Answer

if a sequence of functions converges pointwise to a function and is dominated by an integrable function, then the sequence converges in integral to the integral of the limit function

Answer

if a sequence of functions converges in L^p and is differentiable, then the limit function is differentiable

Answer

if a sequence of functions converges in L^∞ and is continuous, then the limit function is continuous

Question 20

A real-valued function f defined on a set X with a measure μ is essentially bounded if

Accepted Answer

there exists a constant M such that |f(x)| ≤ M for almost all x in X

Answer

f is bounded on a set of finite measure

Answer

f is integrable over X

Question 21

A measure μ on a set X is absolutely continuous with respect to another measure ν if

Accepted Answer

μ(E) = 0 whenever ν(E) = 0

Answer

μ(E) ≤ ν(E) for all measurable sets E

Answer

μ and ν are both σ-finite

Answer

μ(E) = ν(E) for all measurable sets E