Questions related to measures, sigma-algebras, measure spaces, Lebesgue integration and the like.

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29
votes
2answers
3k 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 ...
31
votes
4answers
5k 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 ...
18
votes
2answers
6k 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
2k 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
856 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 ...
32
votes
3answers
3k 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 ...
16
votes
2answers
6k 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 $x$?
26
votes
2answers
7k 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 ...
20
votes
1answer
3k 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
1k 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 ...
25
votes
3answers
3k 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 = \{ ...
10
votes
2answers
1k views

“Scaled $L^p$ norm” and geometric mean

The $L^p$ norm in $\mathbb{R}^n$ is \begin{align} \|x\|_p = \left(\sum_{j=1}^{n} |x_j|^p\right)^{1/p}. \end{align} Playing around with WolframAlpha, I noticed that, if we define the "scaled" $L^p$ ...
12
votes
2answers
2k 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 $\|fg\|_1 \leq C\|g\|_q$ ...
26
votes
4answers
4k 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 ...
11
votes
4answers
646 views

Show $\lim \limits_{n \to \infty} \frac{a_{n+1}}{a_n} = \|f\|_{\infty}$ for $f \in L^{\infty}$

I have a question that I need help with getting started (possibly I would be back for more help). I have a measure space $(X,A,\mu)$ that is finite, and $f \in L^{\infty}(\mu)$. Also, defined is ...
54
votes
4answers
3k 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
2k 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
2k 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? ...
21
votes
5answers
4k views

If $S$ is an infinite $\sigma$ algebra on $X$ then $S$ is not countable

I am going over a tutorial in my real analysis course and there is an exercise that I don't understand a part of his solution. The exercise is: Let $S$ be an infinite $\sigma$ algebra on $X$ .Prove ...
15
votes
2answers
2k 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 ...
23
votes
10answers
3k 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 ...
19
votes
2answers
3k 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 ...
18
votes
2answers
1k 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 ...
19
votes
2answers
4k 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 ...
16
votes
4answers
1k views

Continuous functions are differentiable on a measurable set?

I came across the following challenging problem in my self-study: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be continuous. Then the set of points where $f$ is differentiable is a measurable set. I ...
20
votes
2answers
1k views

Is there a function with infinite integral on every interval?

Could give some examples of nonnegative measurable function $f:\mathbb{R}\to[0,\infty)$, such that its integral over any bounded interval is infinite?
18
votes
2answers
2k 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 $E$?
11
votes
3answers
1k 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 $Y$. ...
6
votes
1answer
554 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
1answer
2k 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. ...
20
votes
9answers
4k 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 ...
18
votes
1answer
2k 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 ...
8
votes
2answers
1k views

Measure of the Cantor set plus the Cantor set

The Sum of 2 sets of Measure zero might well be very large for example, the sum of $x$-axis and $y$-axis, is nothing but the whole plane. Similarly one can ask this question about Cantor sets: If $C$ ...
11
votes
2answers
1k 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 ...
11
votes
1answer
4k 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 ...
5
votes
2answers
248 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
1k 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} ...
7
votes
1answer
1k views

Number of $\sigma$ -Algebra on the finite set

Let $X$ is a nonempty set with $m$ members . How many $\sigma$ -algebra can we make on this set?
9
votes
2answers
2k 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?
7
votes
1answer
957 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 ...
4
votes
2answers
471 views

A Lebesgue measure question [duplicate]

Possible Duplicate: Is there a measurable set $A$ such that $m(A \cap B) = \frac12 m(B)$ for every open set $B$? Is there a measurable set $E \subset [0,1]$ such that for any $0 < a < ...
9
votes
2answers
1k views

Is the image of a null set under a differentiable map always null?

Let $n$ be a positive integer. $\; $ Let $f : \mathbb{R}^n \to \mathbb{R}^n$ be everywhere Frechet differentiable. Let $S$ be a subset of $\mathbb{R}^n$ with Lebesgue measure zero. Does it follow ...
2
votes
1answer
249 views

Prove that two random variables are almost surely equal

$X$ and $Y$ are two random variables such that $Y=E[X|Y]$ almost surely (a.s) and $X= E[Y|X]$ a.s. Prove that $X=Y$ a.s. The hint I was given is to evaluate : $$E[X-Y;X>a,Y\leq a] + E[X-Y;X\leq ...
33
votes
8answers
2k 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 ...
19
votes
2answers
2k 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 ...
10
votes
4answers
862 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 < ...
10
votes
5answers
5k views

Example where union of increasing sigma algebras is not a sigma algebra

If $\mathcal{F}_1 \subset \mathcal{F}_2 \subset \dotsb$ are sigma algebras, what is wrong with claiming that $\cup_i\mathcal{F}_i$ is a sigma algebra? It seems closed under complement since for all ...
12
votes
4answers
1k 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.
8
votes
2answers
2k 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.
12
votes
2answers
1k 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 ...