Questions relating to measures, measure spaces, Lebesgue integration and the like.
20
votes
3answers
938 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 ...
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 $\mathcal{A}$.
...
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 ...
4
votes
1answer
426 views
Limit of $L^p$ norm
Could someone help me prove that given a measure space $(X, \mathcal{M}, \sigma)$ and a measurable function $f:X\to\mathbb{R}$ in $L^{\infty}$ and some $L^{q}$, ...
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
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 ...
12
votes
2answers
979 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 ...
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 ...
15
votes
2answers
771 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 ...
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, ...
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 ...
15
votes
2answers
785 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?
14
votes
2answers
818 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 ...
6
votes
1answer
145 views
Approximating a $\sigma$-algebra by a generating algebra
Theorem. Let $(X,\mathcal B,\mu)$ a finite measure space, where $\mu$ is a positive measure. Let $\mathcal A\subset \mathcal B$ an algebra generating $\cal B$.
Then for all $B\in\cal B$ and ...
5
votes
2answers
600 views
Generalisation of Dominated Convergence Theorem
Wikipedia claims, if $\sigma$-finite the Dominated convergence theorem is still true when pointwise convergence is replaced by convergence in measure, does anyone know where to find a proof of this? ...
11
votes
2answers
844 views
Comparing the Lebesgue measure of an open set and its closure
Let $E$ be an open set in $[0,1]^n$ and $m$ be the Lebesgue measure.
Is it possible that $m(E)\neq m(\bar{E})$, where $\bar{E}$ stands for the closure of $E$?
9
votes
1answer
616 views
If $f$ is measurable and $fg$ is in $L^1$ for all $g \in L^q$, must $f \in L^p$?
Let $f$ be a measurable function on a measure space $X$ and suppose that $fg \in L^1$ for all $g\in L^q$. Must $f$ be in $L^p$, for $p$ the conjugate of $q$? If we assume that $\|fg\|_1 \leq C\|g\|_q$ ...
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 ...
13
votes
1answer
281 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 ...
6
votes
1answer
381 views
Summing over General Functions of Primes and an Application to Prime $\zeta$ Function
Along the lines of thought given here, is it in general possible to substitute a summation over a function $f$ of primes like the following:
$$
\sum_{p\le x}f(p)=\int_2^x f(t) d(\pi(t))\tag{1}
$$
and ...
4
votes
2answers
168 views
Integrate and measure problem.
If $f \in L^{p_0}(X,M,\,u)$ for some $0<p_0 \le\infty$, then
$$1. \lim_{p\to0}\int_{X} |f|^pd\mu=\mu(\{x \in X | f(x) \ne0\}).$$
And if additional assume $\mu(X)=1$,
then I wanna prove that $f ...
11
votes
1answer
518 views
Limit of measures is again a measure
Given a sequence $(\mu_n)_{n\in \mathbb N}$ of finite measures on the measurable space $(\Omega, \mathcal A)$ such that for every $A \in \mathcal A$ the limit
$$\mu(A) = \lim_{n\to \infty} ...
6
votes
2answers
832 views
Does convergence in $L^{p}$ implies convergence almost everywhere?
If I know $\|f_{n}(x) - f(x)\|_{L^{p}(\mathbb{R})} \rightarrow 0$ as $n \rightarrow \infty$, do I know $\lim_{n \rightarrow \infty}f_{n}(x) = f(x)$ for almost every $x$?
3
votes
1answer
1k views
Understanding proof of completeness of $L^{\infty}$
I'm reading page number 4 here. In particular the section where it deals with the case $p=\infty$, that is , showing that $L^{\infty}$ is complete.
...
7
votes
4answers
682 views
Convergence of integrals in $L^p$
Stuck with this problem from Zgymund's book.
Suppose that $f_{n} \rightarrow f$ almost everywhere and that $f_{n}, f \in L^{p}$ where $1<p<\infty$. Assume that $\|f_{n}\|_{p} \leq M < ...
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 ...
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 ...
11
votes
2answers
2k views
$L^p$ and $L^q$ space inclusion
Let $(X, \mathcal B, m)$ be a measure space. For $1 \leq p < q \leq \infty$, under what condition is it true that $L^q(X, \mathcal B, m) \subset L^p(X, \mathcal B, m)$ and what is a counterexample ...
8
votes
1answer
733 views
Proving sets are measurable
The problem statement, all variables and given/known data
The question is from Stein and Shakarchi, Real Analysis 2, Chapter 1, Problem 5:
Suppose $E$ is measurable with $m(E) < \infty$, and ...
6
votes
1answer
498 views
Theorem of Steinhaus
The Steinhaus theorem says that if a set $A \subset \mathbb R^n$ is of positive inner Lebesgue measure then $\operatorname{int}{(A+A)} \neq \emptyset$. Is it true that also ...
1
vote
2answers
292 views
Advantage of accepting non-measurable sets
What would be the advantage of accepting non-measurable sets?
I personally feel that non-measurable sets only exist because of infamous Banach-Tarski paradox...
17
votes
2answers
809 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 ...
11
votes
2answers
815 views
How to split an integral exactly in two parts
This question is a by-product of a conversation with Theo Buehler in comments to this answer. Let's settle definitions.
Definition Let $(\Omega, \mathcal{F}, \mu)$ be a measure space. We say that ...
10
votes
2answers
548 views
Why does a Lipschitz function $f:\mathbb{R}^d\to\mathbb{R}^d$ map measure zero sets to measure zero sets?
Why does a Lipschitz function $f:\mathbb{R}^d\to\mathbb{R}^d$ map measure zero sets to measure zero sets?
It is easy to prove this statement if the domain is bounded. Is there any way to extend ...
4
votes
1answer
527 views
Nearly, but not almost, continuous
Lusin's Theorem asserts that a measurable function f is nearly continuous in the sense that for all $\epsilon>0$ there is a set S of measure less than $\epsilon$ such that f is continuous on the ...
3
votes
2answers
689 views
The construction of a Vitali set
Can anyone explain the concept of a Vitali set? I am not able to understand the construction of the set.
7
votes
1answer
798 views
Steinhaus theorem (sums version)
This is a question from Stromberg related to Steinhaus' Theorem:
If $A$ is a set of positive Lebesgue measure, show that $A + A$ contains an interval.
I can't quite see how to modify the ...
6
votes
2answers
909 views
What is an example of a lambda-system that is not a sigma algebra?
What is an example of a lambda-system that is not a sigma algebra?
5
votes
3answers
867 views
Preimage of generated $\sigma$-algebra
For some collection of sets $A$, let $\sigma(A)$ denote the $\sigma$-algebra generated by $A$.
Let $C$ be some collection of subsets of a set $Y$, and let $f$ be a function from some set $X$ to $Y$. ...
4
votes
2answers
339 views
Showing uniform continuity
I've been trying to show that a function $f$ on the real interval $[a,b]$ which satisfies
$$
f(x)=f(a)+\int_a^xf'(s)\,ds\qquad\text{($f'$ defined almost everywhere)}
$$
must be uniformly continuous ...
9
votes
3answers
697 views
Axiom of choice, non-measurable sets, countable unions
I have been looking through several mathoverflow posts, especially these ones http://mathoverflow.net/questions/32720/non-borel-sets-without-axiom-of-choice , ...
4
votes
3answers
165 views
Lebesgue measure on Riemann integrable function in $\mathbb{R}^2$
As mentioned in a few of my other questions, I am new to measure theory and learning it on my own. I came across an interesting exercise and I would be grateful for/interested in any thoughts.
Setup:
...
25
votes
2answers
684 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 ...
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 ...
8
votes
4answers
632 views
Notation question: Integrating against a measure
Suppose $\mu$ is a measure. Is there any difference in meaning between the notation
$\int f(x)d\mu(x)$
and the notation
$\int f(x) \mu(dx)$?
Many thanks.
7
votes
1answer
1k views
Lebesgue measurable set that is not a Borel measurable set
exact duplicate of Lebesgue measurable but not Borel measurable
BUT! can you please translate Miguel's answer and expand it with a formal proof? I'm totally stuck...
In short: Is there a Lebesgue ...
11
votes
1answer
327 views
Properties of Haar measure
Let $G$ be a locally compact group (but not discrete) and let $m$ be its left Haar measure. Is it true that $\forall \epsilon$ $\exists$ $C$ such that $C$ is a compact neighborhood of the identity and ...
10
votes
2answers
747 views
Uniqueness of product measure (non $\sigma$-finite case)
Let $(X,\mathscr{A},\mu), (Y,\mathscr{B},\nu)$ be two measure spaces, then we have the product measurable space $(X\times Y, \mathscr{A}\times\mathscr{B})$ where $\mathscr{A}\times\mathscr{B}$ is the ...
10
votes
1answer
709 views
Lebesgue measure of the graph of a function
Let $f:R^n \rightarrow R^m$ be any function. Will the graph of f always have Lebesgue measure zero?
1) I could prove that this is true if $f$ is continuous.
2) I suspect it is true if $f$ is ...
2
votes
2answers
696 views
Finding simple, step, and continuous functions to satisfy Lebesgue integral conditions
The problem:
Suppose $E \in \mathfrak{M}$ and that $f$ is (Lebesgue) integrable over $E$. For any $\epsilon > 0$ show that there exist simple, step, and continuous functions $\varphi, \psi, ...
