As a mathematician, having a solid grasp of Lebesgue measure theory is essential for mastering real analysis and succeeding in technical interviews Lebesgue measure provides a more generalized notion of length, area, volume and integration than traditional Riemann integral methods Let’s walk through the key Lebesgue measure concepts and example questions you may face during the interview process.
Developed by French mathematician Henri Lebesgue, Lebesgue measure extends the intuitive concept of measurement to a much wider class of sets compared to the Riemann approach. It handles sets with complex structures and discontinuities that stump Riemann integration.
During interviews, expect questions testing your understanding of Lebesgue measure fundamentals like:
- Measure spaces and measurable sets
- Properties of the Lebesgue measure
- Relationship with Riemann integral
- Lebesgue integration
- Key theorems and applications
I’ll use examples and visuals to explain the core ideas in simple terms. With practice you’ll ace the Lebesgue measure questions and impress the interviewers. Let’s get started!
1. What is the key difference between Riemann and Lebesgue integrals?
The Riemann integral approximates the area under a curve by summing the areas of vertical slices. Lebesgue integration uses horizontal slices of the curve corresponding to the function’s value. This allows handling functions with infinite discontinuities.
Riemann integration struggles with functions that oscillate infinitely within an interval. Lebesgue overcomes this by considering the set of points where the function takes a value.
2. Explain the concept of a measure space.
A measure space (X, Σ, μ) comprises:
- A set X
- A sigma-algebra Σ of subsets of X
- A measure μ that assigns a non-negative value or ∞ to each set in ΣΣ contains X, is closed under complements and countable unions. μ measures the “size” of sets in a way that generalizes length, area etc.
3. What is the Borel sigma-algebra?
The Borel sigma-algebra is the smallest sigma-algebra containing all open sets of a topological space X.
For example, in R, the Borel sets include intervals, countable unions and complements of intervals etc. The Lebesgue measure assigns lengths to Borel sets in a consistent way.
4. When is a set considered Lebesgue measurable?
A set E is Lebesgue measurable iff for every ε > 0, there exist open sets O1 and O2 such that:
E ⊆ O1, O2 ⊆ E and m*(O1 – O2) < ε
where m* denotes Lebesgue outer measure. This measures how close E is to being open.
5. What is the significance of null sets?
A null set has Lebesgue measure zero. This allows disregarding sets like finite points or the rationals when computing integrals. It’s crucial for handling discontinuities.
6. What does it mean for a function to be integrable?
A function f is Lebesgue integrable if:∫|f| < ∞
i.e., the total area under the curve of |f| is finite. This is a more relaxed condition than Riemann integrability.
7. Explain the Lebesgue Dominated Convergence Theorem.
If f_n → f pointwise a.e. and |f_n| ≤ g with ∫g < ∞, then:
- f is Lebesgue integrable
- ∫f = lim ∫f_n
This allows interchanging limits and integrals under domination.
8. What is the Cantor set and its Lebesgue measure?
The Cantor set is constructed by repeatedly removing middle thirds from the interval [0, 1]. This results in removing segments with total length 1, leaving a dust-like set of measure zero.
9. How does Lebesgue measure apply in probability?
Lebesgue measure allows defining probability spaces (Ω, F, P) where the sample space Ω need not be countable. It provides a rigorous way to integrate random variables and compute expectations.
10. Compare the Lebesgue and Dirac delta measures.
The Lebesgue measure generalizes lengths and volumes to more abstract spaces. The Dirac delta is a distribution concentrated at a point. It integrates to 1 but isn’t a true function.δ_0(x) = ∞ if x = 0, 0 otherwise, ∫δ_0 = 1
11. State the Lebesgue Differentiation Theorem.
For f ∈ L^1(R),
f(x) = lim_{h → 0} (1/2h)∫_{x-h}^{x+h} f(t) dt
for almost every x ∈ R. Essentially, the local average around x tends to f(x).
12. Explain absolute continuity of measures.
A measure μ is absolutely continuous with respect to ν if ν(A) = 0 implies μ(A) = 0 for any measurable A. Intuitively, ν dominates μ.
For example, Lebesgue measure dominates counting measure on R.
13. What is the Radon-Nikodym theorem?
If μ << ν, then dμ/dν exists satisfying:
μ(A) = ∫_A (dμ/dν) dν
This allows expressing μ in terms of ν, providing insight into the measures’ relationship.
14. How is Lebesgue integration used in Fourier analysis?
Lebesgue integration allows handling non-smooth functions like absolute value that arise in Fourier analysis. The Lebesgue dominated convergence theorem facilitates term-by-term differentiation and integration of Fourier series.
15. State Fubini’s theorem.
If f(x,y) is integrable on R^2, then:∫∫ f(x,y) dA = ∫ (∫ f(x,y) dx) dy
This allows computing double integrals by iterating the order of integration.
16. What is the Vitali covering lemma?
It states that if E ⊆ R^n is measurable, for any ε > 0, there exist disjoint cubes {Q_i} with diameters at most ε covering E, with Σ vol(Q_i) ≤ (1 + ε)m(E).
This constructs near-optimal covers useful for proving theorems.
17. How can you generate the Lebesgue measure from simpler measures?
Use the monotone convergence theorem. Approximate a set E with a sequence of easily measurable sets {E_n} with E_1 ⊆ E_2 ⊆ … ⊆ E. Then m(E) = lim m(E_n).
18. What is the Caratheodory extension procedure?
It extends a pre-measure m* defined on a ring R (closed under finite unions/intersections) to a full measure m on the sigma-algebra generated by R.
This provides a general measure construction technique.
19. Describe the concept of outer measure.
Outer measure generalizes length to sets not necessarily measurable. The Lebesgue outer measure of a set E is infimum of lengths of covers of E by open sets.
20. How does Lebesgue integration extend Riemann’s?
Lebesgue integration handles a wider class of functions including those with infinite discontinuities. Its integral depends on the function values, not intervals on the real line. This provides a more robust theoretical foundation for integral calculus.
And that concludes our Lebesgue measure interview question rundown! With these concepts handy, you’ll confidently tackle the tough theoretical questions on your upcoming interviews. Wishing you the very best.
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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. When you need Lebesgue for a new project, why do you ask this? You might be 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. “Why should I learn about real numbers when I already know about rationals?” is a question I ask my students. 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?
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.
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.
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.
Is the Lebesgue measure of an interval equal to its length?
In Theorem 2.1.3 we show that, as expected, the Lebesgue measure of an interval is indeed equal to its length. 5 6 2. Lebesgue Measure Sets will be “measured” by approximating them by countable unions of intervals. We will approximate our sets “from above,” i.e., we will consider unions of intervalscontainingour sets.
