Questions relating to measures, measure spaces, Lebesgue integration and the like.
41
votes
3answers
1k views
Is it possible for a function to be in $L^p$ for only one $p$?
I'm wondering if it's possible for a function to be an $L^p$ space for only one value of $p \in [1,\infty)$ (on either a bounded domain or an unbounded domain).
One can use interpolation to show that ...
28
votes
7answers
1k views
Why do we restrict the definition of Lebesgue Integrability?
The function $f(x) = \sin(x)/x$ is Riemann Integrable from $0$ to $\infty$, but it is not Lebesgue Integrable on that same interval. (Note, it is not absolutely Riemann Integrable.)
Why is it we ...
27
votes
5answers
1k views
False beliefs about Lebesgue measure on $\mathbb{R}$
I'm trying to develop intuition about Lebesgue measure on $\mathbb{R}$ and I'd like to build a list of false beliefs about it, for example: every set is measurable, every set of measure zero is ...
26
votes
4answers
417 views
To show that the set point distant by 1 of a compact set has Lebesgue measure $0$
Could any one tell me how to solve this one?
Let $K$ be a compact subset of $\mathbb{R}^n$, and $$A:=\{x\in\mathbb{R}^n:d(x,K)=1\}.$$
Show that $A$ has Lebesgue measure $0$.
Thank you!
25
votes
2answers
693 views
Integration of forms and integration on a measure space
In Terence Tao's PCM article: DIFFERENTIAL FORMS AND INTEGRATION, it is pointed out that there are three concepts of integration which appear in the subject(single-variable calculus):
the ...
23
votes
3answers
837 views
Random Variable Inequality
Doing a little reading over the break (The Probabilistic Method by Alon and Spencer); can't come up with the solution for this seemingly simple (and perhaps even a little surprising?) result:
(A-S ...
21
votes
4answers
666 views
Correspondences between Borel algebras and topological spaces
Though tangentially related to another post on MathOverflow (here), the questions below are mainly out of curiosity. They may be very-well known ones with very well-known answers, but...
Suppose ...
20
votes
3answers
944 views
Set of continuity points of a real function
I have a question about subsets $$
A \subseteq \mathbb R
$$
for which there exists a function $$f : \mathbb R \to \mathbb R$$ such that the set of continuity points of $f$ is $A$. Can I characterize ...
20
votes
2answers
710 views
If a continuous function is positive on the rationals, is it positive almost everywhere?
I made up this question, but unable to solve it:
Let $f : \mathbb R \to \mathbb R$ be a continuous function such that $f(x) > 0$ for all $x \in \mathbb Q$. Is it necessary that $f(x) > 0$ ...
20
votes
3answers
911 views
Why is the Daniell integral not so popular?
The Riemann integral is the most common integral in use and is the first integral I was taught to use. After doing some more advanced analysis it becomes clear that the Riemann integral has some ...
20
votes
3answers
2k views
How do people apply the Lebesgue integration theory?
This question has puzzled me for a long time. It may be too vague to ask here. I hope I can narrow down the question well so that one can offer some ideas.
In a lot of calculus textbooks, there is ...
19
votes
1answer
458 views
Universally measurable sets of $\mathbb{R}^2$
$$\text{Is }{{\cal B}(\mathbb{R}^2})^u={{\cal B}(\mathbb{R}})^u\times {{\cal B}(\mathbb{R}})^u\,?\tag1$$
Is the $\sigma$-algebra of universally measurable sets on $\mathbb{R}^2$ equal to the
product ...
18
votes
10answers
1k views
Seeking a layman's guide to Measure Theory
I would like to teach myself measure theory. Unfortunately most of the books that I've come across are very difficult and are quick to get into Lemmas and proofs. Can someone please recommend a ...
18
votes
1answer
424 views
What are some good intuitions for understanding Souslin's operation $\mathcal{A}$?
What are some good intuitions for understanding Souslin's operation $\mathcal{A}$?
Recall the definition: Let $S = \mathbb{N^{<N}} = \bigcup_{n = 1}^\infty \mathbb{N}^n$ be the set of ...
17
votes
4answers
954 views
Tricks to remember Fatou's lemma
For a sequence of non-negative measurable functions $f_n$, Fatou's lemma is a statement about the inequality
$\int \liminf_{n\rightarrow \infty} f_n \mathrm{d}\mu \leq \liminf_{n\rightarrow ...
17
votes
4answers
515 views
Is there a $\sigma$-algebra on $\mathbb{R}$ strictly between the Borel and Lebesgue algebras?
So, after proving that $\mathfrak{B}(\mathbb{R})\subset \mathfrak{L}(\mathbb{R})$, I asked myself, and now asking you, is there a set $\mathfrak{S}(\mathbb{R})$, which satisfies:
...
17
votes
3answers
2k views
Is every Lebesgue measurable function on $\mathbb{R}$ the pointwise limit of continuous functions?
I know that if $f$ is a Lebesgue measurable function on $[a,b]$ then there exists a continuous function $g$ such that $|f(x)-g(x)|< \epsilon$ for all $x\in [a,b]\setminus P$ where the measure of ...
17
votes
3answers
1k views
The set of differences for a set of positive Lebesgue measure
Quite a while ago, I heard about a statement in measure theory, that goes as follows:
Let $A \subset \mathbb R^n$ be a Lebesgue-measurable set of positive measure. Then we follow that $A-A = \{ ...
17
votes
2answers
811 views
Two definitions of Lebesgue integration
Normally, Lebesgue integral, for positive measures, is defined in the following way. First, one defines the integral for indicator functions, and linearly extend to simple functions. Then, for a ...
17
votes
1answer
589 views
Do inequalities that hold for infinite sums hold for integrals too?
Let $\mathbb{R}_{\geq0}$ denote the set of non-negative reals and $+\infty$, and $\mathbb{Z}^+$ denote the set of positive integers. I will also let $\lambda$ denote the Lebesgue measure on ...
16
votes
3answers
2k views
Construction of a Borel set with positive but not full measure in each interval
I was wondering how one can construct a Borel set that doesn't have full measure on any interval of the real line but does have positive measure everywhere.
To be precise, if $\mu$ denotes Lebesgue ...
16
votes
5answers
264 views
Examples of properties that hold almost everywhere, but that explicit examples unknown.
In measure theory one makes rigorous the concept of something holding "almost everywhere" or "almost surely", meaning the set on which the property fails has measure zero.
I think it is very ...
16
votes
2answers
1k views
The union of a strictly increasing sequence of $\sigma$-algebras is not a $\sigma$-algebra
The union of a sequence of $\sigma$-algebras need not be a $\sigma$-algebra, but how do I prove the stronger statement below?
Let $\mathcal{F}_n$ be a sequence of
$\sigma$-algebras. If the ...
16
votes
1answer
652 views
$\sigma$-algebra in Riesz representation theorem
Let $X$ be a locally compact Hausdorff space and $I$ - a positive linear functional on $C_c(X)$. Then according to the Riesz representation theorem
there exist a $\sigma$-algebra $\mathfrak{M}$ in $X$ ...
15
votes
2answers
791 views
examples of measurable functions on $\mathbb{R}$
Could give some examples of nonnegative measurable function $f:\mathbb{R}\to[0,\infty)$, such that its integral over any bounded interval is infinite?
15
votes
2answers
553 views
Clarifying the relationship between outer measures, measures and measurable spaces: the converse direction
This is related to my measure theory class, but it's not homework. The motivation behind this post is to understand the exact relationship between the three objects mentioned in the title.
...
15
votes
1answer
1k views
Cardinality of Borel sigma algebra
It seems it's well known that if a sigma algebra is generated by countably many sets, then the cardinality of it is either finite or $c$ (the cardinality of continuum). But it seems hard to prove it, ...
15
votes
2answers
774 views
Infinite product of measurable spaces
Suppose there is a family (can be
infinite) of measurable spaces. What
are the usual ways to define a sigma
algebra on their Cartesian product?
There is one way in the context of
defining product ...
14
votes
8answers
1k views
Reference book on measure theory
I post this question with some personal specifications. I hope it does not overlap with old posted questions.
Recently I strongly feel that I have to review the knowledge of measure theory for the ...
14
votes
3answers
317 views
Locally non-enumerable dense subsets of R
Today after lunch I was hungry for math problems so I started begging for some at the department and finally someone threw me this: Can $\mathbb{R}$ be partitioned into two non-countable dense ...
14
votes
3answers
396 views
Weird subfields of $\Bbb{R}$
I found this problem, and I can't get an answer to it:
Prove that there are subfields of $\Bbb{R}$ that are
a) non-measurable.
b) of measure zero and continuum cardinality.
I can't ...
14
votes
2answers
822 views
The sum of an uncountable number of positive numbers
Claim:If $(x_\alpha)_{\alpha\in A}$ is a collection of real numbers $x_\alpha\in [0,\infty]$
such that $\sum_{\alpha\in A}x_\alpha<\infty$, then $x_\alpha=0$ for all but at most countably many ...
14
votes
1answer
321 views
Kakutani skyscraper is infinite
Karl E. Petersen's book "Ergodic Theory", chapter 2, exercise 9, on page 56
Prove that for any ergodic measure preserving transformation $T:X\rightarrow X$ on a non-atomic probability space $(X, ...
14
votes
2answers
408 views
Pointwise existence of Radon-Nikodým derivative sufficient for absolute continuity?
Let $\mu$ be a finite, nonatomic, signed Radon measure on $([0,1],\mathcal{B}([0,1]))$. For all $x \in ]0,1[$ the limit
$$\lim_{\varepsilon \to 0} ...
14
votes
1answer
445 views
Integral of measurable spaces
If for each $t\in I=[0,1]$ I have a measurable space $(X_t,\Sigma_t)$, is there a standard notion which will give a measurable space deserving to be called the integral $\int_I X_t\,\mathrm d t$?
...
13
votes
3answers
1k views
The $\sigma$-algebra of subsets of $X$ generated by a set $\mathcal{A}$ is the smallest sigma algebra including $\mathcal{A}$
I am struggling to understand why it should be that the $\sigma$-algebra of subsets of $X$ generated by $\mathcal{A}$ should be the smallest $\sigma$-algebra of subsets of $X$ including $\mathcal{A}$.
...
13
votes
1answer
526 views
Medial Limit of Mokobodzki (case of Banach Limit)
A classical Banach limit is very usefull concept for me, but there is a problem with the integration and even with the measurability, this means for a sequence $(f_n)_{n\in \mathbb{N}}$ of measurable ...
13
votes
1answer
282 views
On the equality case of the Hölder and Minkowski inequalites
I'm following the book Measure and Integral of Richard L. Wheeden and Antoni Zygmund. This is the problem 4 of chapter 8.
Consider $E\subseteq \mathbb{R}^n$ a measurable set. In the following all the ...
13
votes
2answers
2k views
Lebesgue measurable but not Borel measurable
I'm trying to find a set which is Lebesgue measurable but not Borel measurable.
So I was thinking of taking a Lebesgue set of measure zero and intersecting it with something so that the result is not ...
13
votes
2answers
428 views
How badly can Dini's theorem fail if the p.w. limit isn't continuous?
Dini's theorem is commonly seen in real analysis courses (possibly with the requirement that $X$ be a compact metric space if topological spaces are still off in the future), but suppose one wanted to ...
13
votes
1answer
309 views
Controlling the Size of an Open Cover of a Set of Measure Zero
Suppose we have a subset $A\subset\mathbb{R}$ of Lebesgue measure zero contained in a compact interval, say $[0,1]$. We know that since $A$ has measure zero we can cover $A$ with a countable set of ...
13
votes
2answers
381 views
Question about the Riesz representation theorem(s)
I am looking at two seemingly same (but not quite) Riesz representation theorems:
(Wikipedia) Let $X$ be a locally compact Hausdorff space. Let $C_c(X)$ be the space of compactly supported continuous ...
13
votes
1answer
210 views
Measurable subset of $\mathbb{R}$ with a specific property
Let $A$ be a subset of $\mathbb{R}$ such that its intersection with every finite segment is Lebesgue measurable. I am looking for an example of such an $A$ with the additional property that the ...
13
votes
1answer
942 views
Geometric Explanation of Tamagawa Numbers
Sometimes in order to understand a concept thoroughly we need to have a algebraic view ( in terms of equations ) and corresponding geometric view.
My interest always lies with understanding the ...
13
votes
0answers
174 views
Measure theoretic definition of curl
Is there a good measure theoretic definition of curl?
To give an idea of the sort of equation that I'm looking for, here's now I define grad and div. For the gradient, say we are given a Fréchet ...
12
votes
2answers
426 views
Big Rudin Exercise 3.26 - Which integral is larger
This is exercise 3.26 in Rudin's Real & Complex Analysis:
If $f$ is a positive measurable function on $[0,1]$, which is larger,
$$\int_0^1 f(x) \log f(x) \, dx$$
or
$$\int_0^1 f(s) \, ds ...
12
votes
2answers
203 views
Finding $\displaystyle \lim_{n \to \infty} \int_0^\infty \frac{e^{-x}\cos{x}}{nx^2 + \frac{1}{n}}dx$
Prove that $\displaystyle \lim_{n \to \infty} \int_0^\infty \frac{e^{-x}\cos{x}}{nx^2 + \frac{1}{n}}dx$ exists and determine its value.
12
votes
2answers
981 views
Vitali-type set with given outer measure
Is it possible to construct a non-measurable set in $[0,1]$ of a given outer measure $x \in [0,1]$? This will probably require the axiom of choice. Does anyone have a suggestion?
Edit: I forgot to ...
12
votes
1answer
1k views
Example of a Borel set that is neither $F_\sigma$ nor $G_\delta$
I'm looking for subset $A$ of $\mathbb R$ such that $A$ is a Borel set but $A$ is neither $F_\sigma$ nor $G_\delta$.
12
votes
3answers
386 views
Why does the Continuum Hypothesis make an ideal measure on $\mathbb R$ impossible?
On the page 43 of Real Analysis by H.L. Royden (1st Edition) we read: "(Ideally) we should like $m$ (the measure function) to have the following properties:
$m(E)$ is defined for each subset $E$ of ...
