The Lebesgue measure is one of the most important ideas in real analysis and measure theory. It is also very important for understanding probability theory, integration, and many other areas of mathematics. This guide gives you a complete rundown of typical Lebesgue measure interview questions, along with helpful answers and explanations to help you get ready for your next interview.
1. What is the Lebesgue measure?
The Lebesgue measure, denoted by $lambda$, is a way of assigning a “size” or “length” to sets in $mathbb{R}^n$. It extends the concept of length from intervals to more general sets, including those with intricate geometric structures or even fractal boundaries. Unlike the naive notion of length, the Lebesgue measure can handle sets that are “too irregular” to be assigned a length in the traditional sense.
2. What are the properties of the Lebesgue measure?
The Lebesgue measure possesses several key properties including
- Translation invariance: The measure of a set does not change if it is shifted by a constant vector.
- Scaling invariance: The measure of a set scales proportionally with its size.
- Countable additivity: The measure of a countable union of disjoint sets is equal to the sum of the measures of the individual sets.
- Completeness: Every set that differs from a Lebesgue measurable set by a set of measure zero is also Lebesgue measurable.
3. How is the Lebesgue measure defined?
The Lebesgue measure is defined using the concept of outer measure. The outer measure of a set is the infimum of the sum of the measures of open sets that cover the set. The Lebesgue measure of a set is then defined as the outer measure of the set if it coincides with the outer measure of its complement.
4. What are some examples of sets that are not Lebesgue measurable?
There exist sets that are not Lebesgue measurable, such as the Vitali set. These sets are “too irregular” to be assigned a Lebesgue measure.
5. What is the relationship between the Lebesgue measure and the Riemann integral?
The Lebesgue measure provides a foundation for the Lebesgue integral which is a generalization of the Riemann integral. The Lebesgue integral can integrate functions that are not Riemann integrable such as functions with discontinuities or unbounded variations.
6, How is the Lebesgue measure used in probability theory?
The Lebesgue measure is used to define probability measures on $mathbb{R}^n$. A probability measure assigns a probability to each event, which is a set of outcomes. The Lebesgue measure allows us to define probabilities for events involving continuous random variables.
7. What are some applications of the Lebesgue measure in other areas of mathematics?
The Lebesgue measure finds applications in various areas of mathematics, including:
- Harmonic analysis: The Lebesgue measure is used to study the properties of functions and operators related to Fourier analysis.
- Geometric measure theory: The Lebesgue measure is used to define and study geometric objects such as sets of finite perimeter and fractal sets.
- Ergodic theory: The Lebesgue measure is used to study dynamical systems and their long-term behavior.
8 What are some common challenges associated with the Lebesgue measure?
One challenge associated with the Lebesgue measure is its abstractness. The definition of the Lebesgue measure using outer measure can be challenging to grasp intuitively. Another challenge is that the Lebesgue measure is not always easy to compute explicitly for complex sets.
9. How can I improve my understanding of the Lebesgue measure?
To improve your understanding of the Lebesgue measure, you can:
- Study the formal definition and properties of the Lebesgue measure.
- Work through examples and exercises involving the Lebesgue measure.
- Read books and articles on real analysis and measure theory.
- Discuss the Lebesgue measure with other mathematicians or attend lectures and workshops.
10. Can you provide some additional resources for learning about the Lebesgue measure?
Here are some additional resources that you may find helpful:
- Books:
- “Real Analysis” by Royden and Fitzpatrick
- “Measure and Integration” by Stein and Shakarchi
- “Introduction to Real Analysis” by Bartle and Sherbert
- Websites:
- MathWorld: Lebesgue Measure
- Wikipedia: Lebesgue Measure
- Khan Academy: Lebesgue Measure
- Videos:
- 3Blue1Brown: What is the Lebesgue Measure?
- Numberphile: The Lebesgue Measure
The Lebesgue measure is a powerful tool in real analysis and measure theory. By understanding its properties and applications, you can gain a deeper understanding of integration, probability theory, and many other areas of mathematics.
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6 Answers 6 Sorted by:
This was American Mathematical Monthly Problem #11526, circa 2010:
Proof. (Mouse over below…)
The existence of normal numbers.
I liked the following example.
Theorem: Let R be a rectangle in the plane with sides parallel to the axes. Let’s say that R is split into a finite number of smaller rectangles whose sides are all parallel to the axes and at least one of their lengths is an integer. Then the big rectangle R also has the same property.
Proof: Consider the complex measure $dmu =dxdye^{2pi i (x+y)}$ on the plane. The hypotheses imply that each of the smaller rectangles has $mu$ measure zero. By summing up, the big rectangle R also has $mu$ measure zero.
This isnt a direct answer, but it may be on topic, depending on the motivation for the question.
I feel like I owe it to my students to explain why they need to learn the Lebesgue integral when they already know the Riemann integral when I teach measure theory. What is the new use for Lebesgue? Maybe you’re asking this because you’re having the same problem.
I guess you can cook up examples of functions that are Lebesgue but not Riemann integrable. But that is going to appear contrived. I tell the class that it’s like asking, “Why should I learn about real numbers if I already know about rationals?” One answer is that there are examples of infinite series that don’t have a sum in $mathbb{Q}$ but do in $mathbb{R}$. A more formal way to say this is that $mathbb{R}$ is complete as a metric space. This has so many benefits that it is clearly worth mentioning.
You can think of the change from Riemann to Lebesgue as a “completion” in the same way that $mathbb{R}$ is a completion of $mathbb{Q}$. The set of Riemann integrable functions isn’t complete if you use $d(f,g) = int_0^1 |f – g|, dx$ to measure the distance between two functions on $[0,1]$. When you finish it, you get the space of Lebesgue integrable functions (minus functions that disappear off a null set, but we’ll talk about that later).
1) Averaging tricks of all kinds, e. g., the entire area of integral geometry. For a specific simple example, see Crofton formula and its consequences listed in the linked article. A version of the formula can be used to prove that if a curve on the sphere is not contained in any hemisphere, then its length is at least $2pi r$, and there are many more serious application too. To answer a possible objection, even for smooth curves the integrand is neither continuous nor bounded; good luck working out the proof with a weaker version of the integral.
Of a similar spirit is e. g. Weyls unitarian trick.
2) The Lebesgue measure makes it easy to create an endless chain of random variables that are all independent and have known distributions. This is (kind of) important for Probability Indeed, binary digits on $[0,1]$ give you an infinite sequence of i. i. d. Bernoullis variables. Rearranging them, you get an infinite sequence of independent infinite sequences of i. i. d. Bernoullis, which is the same as and infinite sequence of uniform random variables on $[0,1]$. Post-composing with functions gives arbitrary distributions.
3) A good theory of integration is needed for completeness and duality in $L^p$ spaces, which have many uses: to show that a function exists, all you have to do is define it or create a Cauchy sequence that goes with it. For a simple example, see this answer or the $L^2$ projection proof of conditional expectation in Williams’ book.
When you give another answer, I don’t know what to do; this has nothing to do with my last example. The most spectacular applications of measure theory that I know come from Margulis work. For example, suppose $Gamma subset SL_3(mathbb R)$ is a discrete subgroup with compact quotient. Then Margulis shows that every non-trivial normal subgroup of $Gamma $ has finite index in $Gamma$. The proof uses measure theory ( and a lot else besides) in a serious way. His proof that such a $Gamma$ is arithmetic also uses ergodic theory (and measure theory). These purely “algebraic” statements were proved by use of measure theory.
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A horizontal integral?! Introduction to Lebesgue Integration
FAQ
What is Lebesgue measure used for?
What is the intuition behind Lebesgue measure?
How do you check if a set is Lebesgue measurable?
What is the uniqueness of Lebesgue measure?
How do you know if a set is Lebesgue measurable?
De nition 3.1. In general, a set is Lebesgue measurable if, for some > 0, there exists and open set G such that E G and jG Eje < , where E Rn is the set we desire to measure. If E is measurable, we may then de ne jEj = jEje, the Lebesgue measure. Two important consequences that characterise the Lebesgue measure are listed below.
What are Lebesgue measurable sets?
These sets are, by definition, the Lebesgue-measurable (briefly L L -measurable) sets; m∗ m ∗ and m m so defined are the ( n n -dimensional) Lebesgue outer measure and Lebesgue measure. Lebesgue premeasure v v is σ σ -additive on C, C, the intervals in En E n.
Is a Lebesgue measurable function?
where ϕ is a Lebesgue measurable function, and the domain of the function is partitioned into sets S₁, S₂, …, Sₙ, m (Sᵢ) is the measure of the set Sᵢ. And a₁, a₂, …, aₙ are in [0, ∞]. Using the Lebesgue approach, the given function is approximated using a simple function. The ϕ presented in Equation 2.1 is such a simple function.
How to understand the Lebesgue measure?
To really grasp an intuitive understanding of the Lebesgue Measure and some of its applications, a few fundamental de nitions and theorems regarding Real Analysis need to be established. Additionally, these theorems and de nitions will assist in elucidating the notation that will be employed henceforth. De nition 1.1.
