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

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2
votes
2answers
293 views

Convergence of Lebesgue integrals

I am sitting on this multiple-choice question and I cannot answer it, nor say if it is right or wrong: Given non-negative, Lebesgue-integrable functions $f,f_k\colon E\rightarrow \mathbb{R}^+$ with ...
21
votes
4answers
820 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 ...
2
votes
1answer
96 views

Description of subspace in $L^2(\mathbb{T})$ of functions with vanishing positive Fourier coefficients

Denote, by $$ H^2= \left\{f\in L^2(\mathbb{T}): n\in\mathbb{N} \rightarrow \int \limits_{\mathbb{T}} f(z)z^nd\mu(z)=0 \right\} $$ $$ \Vert ...
4
votes
1answer
314 views

Function such that its square is not integrable

I have some set $A$ of Lebesgue measure $\mu(A)=1$. Does this imply that there is some measurable function $f: \mathbb{R}^n \to \mathbb{R}$ such that $$\int_A |f| d\mu< \infty, \int_A |f|^2 d\mu= ...
6
votes
1answer
1k views

The subset of non-measurable set

If $A$ is a non-measurable set in $\mathbb R^n$ (in the sense of Lebesgue), does it necessarily contain a positive measurable subset?
2
votes
1answer
362 views

How is an integral with respect to a Hausdorff measure defined?

In a reply by Corey: For integrals of scalar-valued functions on unoriented subsets of $\mathbb{R}^n$, one can use the Lebesgue integral with respect to $k$-dimensional Hausdorff measure ...
6
votes
2answers
302 views

Counterexample to $f_ng_n \not\to fg$ in measure

I'm looking for a pair of sequences $f_n \to f$ in measure, $g_n \to g$ in measure, where $f_ng_n \not\!\to fg$ in measure. I've tried a number of things with characteristic functions that move ...
7
votes
1answer
502 views

Weird measurable set

In the following, consider the Lebegue measure in $\mathbb{R}^d$. Consider $E\subseteq \mathbb{R}^d$ measurable, with $0\lt m(E)\lt\infty$, such that any measurable subset $F$ of $E$ satisfies ...
9
votes
1answer
644 views

Application of Dominated Convergence Theorem?

Find with proof the following limit: $$\lim_{n \to \infty} \int_{-\infty}^{\infty} \frac{(\sin(x))^n}{x^2}dx$$ I want to use the DCT but I cannot seem to dominate $f_{n}(x)=\frac{(\sin(x))^n}{x^2}$ ...
2
votes
2answers
239 views

reference for “compactness” coming from topology of convergence in measure

I have found this sentence in a paper of F. Delbaen and W. Schachermayer with the title: A compactness principle for bounded sequences of martingales with applications. (can be found here) On page 2, ...
14
votes
2answers
725 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 ...
2
votes
1answer
185 views

Continuity and $L^p$ spaces

I have been wondering how to solve this question I saw in a textbook. Given $ g \in \bigcup _{1\leq p\leq \infty} L^{p}$ define, for $ r \in [ 0,1]$ , $$ G(r) = \int_{0}^{r} g(t) dt \;.$$ Show that ...
4
votes
2answers
252 views

$f = 0$ outside a set of measure zero implies $\int_a^b f \, dx= 0$

Let $f \colon [a,b] \to \mathbb R$ bounded, such that $f(x) = 0$ for every $x \in [a,b]$ except in a set $J$ of measure zero. When we say that a set $J$ has measure zero, if given any $\varepsilon ...
6
votes
4answers
517 views

Convergence in metric and a.e

How might I show that there's no metric on the space of measurable functions on $([0,1],\mathrm{Lebesgue})$ such that a sequence of functions converges a.e. iff the sequence converges in the metric?
1
vote
0answers
72 views

Existence of measure factorization

Let $(\Omega,\mathcal F,\mathbb P)$ a probability space, $(X,\mathcal B)$ a measurable space and $m$ a probability measure on $\Omega\times X$ such that its projection on $\Omega $ is equal to ...
4
votes
1answer
236 views

Computing $\lim\limits_{n\to\infty}\int_X ~{\left\{\cos\left(\pi f(x)\right) \right\}}^{2n}~\text{d}\mu(x)$

Let $(X,M,\mu)$ be a finite measure space and let $f$ be a real-valued and measurable function on $X$. How do I evaluate $$ \lim_{n\rightarrow \infty} \int_X ~{\left\{\cos\left(\pi f(x)\right) ...
2
votes
2answers
679 views

How to show that a sequence of integrable functions converges to a function in $L^1$-norm

I'm new and I have a question I need help in solving. This isn't homework. The question is as follows: Let $(X,M,\mu)$ be a finite measure space and suppose $f_n$ is a sequence of integrable ...
0
votes
1answer
140 views

Prove series is unbounded on any interval

Let $f(x) = x^{-1/2}$ for $ x \in (0,1)$ and zero otherwise. Let $F(x) = \sum 2^{-n} f(x - r_{n})$ where $ {r_{n}} $ is an enumeration of rationals. 1) Prove that $F(x)$ is integrable and thus its ...
2
votes
2answers
233 views

Proving integral = infinity

Let $F$ be a closed set of $ \mathbb{R} $ whose complement has finite measure. Let $\delta(x) = d (x, F) =\inf \{ |x - z| \mid z \in F\}$. Prove $ \delta$ continuous by proving $| \delta(x) - ...
7
votes
4answers
306 views

showing that $g=0$ almost everywhere on $[0,1]$

Let $g \in L^1[0,1]$. Suppose that given any pair of rationals $0\leq p\lt q \leq 1$, we have $$\int_p^q g(x) d\mu=0.$$ Please I would like help in showing that $g=0$ almost everywhere on $[0,1]$.
13
votes
1answer
1k 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 ...
3
votes
0answers
359 views

Measure defined on Semi-Algebra and on Algebra

Let $\Omega$ be a set, $\mathcal B$ is a semi-algebra that contains $\Omega$, and let $\mu \colon \mathcal B\rightarrow [0,\infty]$ be a measure defined on $(\Omega,B)$. Now define the algebra ...
2
votes
2answers
779 views

Integrable function $f$ on $\mathbb R$ does not imply that limit $f(x)$ is zero

1) Construct a continuous function $f$ on $\mathbb{R}$ that is integrable on $\mathbb{R}$ but $\displaystyle\limsup_{x \to \infty} f(x) = \infty$. I took the function that is equal to $n$ on $[n, n ...
0
votes
1answer
119 views

Criterion for a set to be measurable

Okay. So I know that a set $E$ is called measurable if for any set $A$, we have $$ m^\ast(A)=m^\ast(A\cap E)+m^\ast(A\cap E^c).$$ Recently, I came across a Lemma which says that a set $A\subset ...
4
votes
2answers
633 views

Show that $\int_{-\pi}^\pi ~f(x) \cos (nx) \mathrm{d}\mu(x)$ converges to $0$

I need some help on the following problem. Let $f\in L_1([-\pi,\pi])$. Then $\int_{-\pi}^\pi ~f(x) \cos (nx) \mathrm{d}\mu(x) \to 0$, where $\mu$ is the Lebesgue measure on $[\pi,\pi]$. Any ...
2
votes
2answers
501 views

proving a function satisfies a Lipschitz condition

Let $F$ be a closed set of $ \mathbb{R} $ whose complement has finite measure. Let $\delta(x) = d (x, F) =\inf \{ |x - z| \mid z \in F\}$. Prove $ \delta$ continuous by proving $| \delta(x) - ...
8
votes
2answers
561 views

A sequence of measures on a sigma algebra

Let $X$ be a set and $\mathcal{A}$, a sigma algebra of subsets of $X$. Let $\{\mu_n\}$ be a sequence of measures of $\mathcal{A}$ such that $\mu_{n+1}(E)\geqslant \mu_n(E)$ for every $E\in ...
2
votes
1answer
150 views

Existence of a structure-preserving mapping between two spaces?

I have some questions, but not sure if they are meaningful: Suppose $X$ and $Y$ are two arbitrary measurable spaces. Does there exist a measurable mapping from $X$ to $Y$? Suppose $X$ and $Y$ are ...
1
vote
0answers
80 views

To construct an invariant measure on the product space

$M$ is a separable complete metric space. $N$ is a compact metric space. $F$, defined by $F(x,y)=(f(x),g(x,y))$, is a continuous transformation from $M\times N$ to itself. We know there is a measure ...
3
votes
0answers
149 views

Jordan decomposition for a function on an interval

Let $f:[a,b] \rightarrow \mathbb R$ be of bounded variation. Let $V(f)_a ^x$ denote its total variation from $a$ to $x$. Define $f_1(x) = \frac{V(f)_a ^x + f(x)}{2}$ and $f_2(x) = \frac{V(f)_a ^x ...
12
votes
2answers
2k views

On Lipschitz condition and absolute continuity

A function $f(x)$ on $[0,1]$ is said to satisfy a Lipschitz condition if there exists a constant $M$, such that $$|f(x)-f(y)|\leqslant M|x-y| ~\forall~x,y\in[0,1]. $$ I want to show the ...
0
votes
2answers
413 views

Existence of a bounded measurable function

How do I prove the following: Let $f$ be a measurable function on $[0,1]$ such that $f$ is finite almost everywhere. Then for any $\varepsilon \gt 0$, $\exists$ a bounded measurable function $g$ ...
6
votes
1answer
378 views

Do these $\sigma$-algebras on second countable spaces coincide?

There's an interesting property that if $(X,\mathcal{T})$ and $(Y,\mathcal{S})$ are topological spaces, then the Borel $\sigma$-algebra of $X\times Y$ with the product topology includes the product ...
3
votes
1answer
178 views

If the Fourier transform of a signed measure is identically zero, is the same true of the measure?

I am trying to prove the following seemingly obvious fact: Let $\mu$ be a finite signed measure on $\mathbb R$. Suppose that $\hat\mu(u) = \int_\mathbb R e^{iux} d\mu(x) = 0$ for all $u$. Then ...
1
vote
1answer
61 views

Making a function out of several measurable functions

This is something I've been curious about. Suppose $(X,\mathcal{R})$ is some measurable space, and $X=\bigcup_n A_n$ where the $A_n$ are measurable, but not necessarily disjoint. On each of these ...
2
votes
1answer
516 views

Importance of a result in measure theory

Let's consider the following result. There exists a borel set $A\subset [0,1]$ such that $0<m(A\cap I)<m(I)$ for every subinterval $I$ of $[0,1]$, where $m$ is Lebesgue measure. Can one ...
4
votes
1answer
829 views

Does the set of differences of a Lebesgue measurable set contains elements of at most a certain length?

I want to show that if $E\subset \mathbb{R}^n$ is a Lebesgue measurable set where $\lambda(E)>0$, then $E-E=\{x-y:x,y\in E\}\supseteq\{z\in\mathbb{R}^n:|z|<\delta\}$ for some $\delta>0$, ...
7
votes
1answer
2k views

A simpler proof of Jensen's inequality

Jensen's inequality states that if $(X,\mu)$ is a measure space with $\mu(X) = 1$, $\phi$ is convex, and $f:X \rightarrow \mathbb R$ is integrable, then $$\phi\left(\int fd\mu\right) \leq \int \phi ...
5
votes
2answers
652 views

Dominated Convergence Theorem using Egorov

I've been reading on Dominated Convergence Theorem and its proof using Fatou-Lebesgue, but I can't seem to figure out how to do so with Egorov's theorem. If $\nu$ is a finite Baire measure on a ...
1
vote
1answer
437 views

Exception to monotone and dominated convergence theorem

If we consider the space $L^\infty ([0,1],\,dx)$, why is it that both monotone and dominated convergence fail? My first take on the problem was to consider the characteristic function $f_n$ of ...
1
vote
1answer
192 views

$L^p$ spaces in integration measure

This question looks simple at the first glance but ... I have tried to combine the theorems and definitions on $L^p$ spaces to solve this question but I have not been able to do so. I need help to ...
4
votes
2answers
450 views

Product of two probability kernel is a probability kernel?

Let $ (\mathbb{X} _i, \mathscr{X}_i) $ and $ (\mathbb {Y} _i, \mathscr {Y} _i) $ measurable spaces with $ i = 1, 2 $. Let $ \gamma_i: \mathscr {X}_i\times\mathbb {Y}_i\longrightarrow [0,1] $ a ...
1
vote
2answers
339 views

Measure Product Theorem: may non-$\sigma$-finiteness result unique product?

Let $i\in\{1,2\}$. The Measure Product Theorem states that, given the measure spaces $(X_i,\Sigma_i,\mu_i)$, there is at least one product measure $\pi$ such that $\pi(A_1\times ...
1
vote
2answers
171 views

Speed of convergence of Lebesgue integrable function as domain goes to zero

For $p \in [1, \infty]$ find values of $\lambda$ such that $\lim\limits_{\epsilon \to 0^{+}} \frac{1}{\epsilon^{\lambda}} \int_{0}^{\epsilon} f = 0$ for all $f \in L^{p}[0, 1]$. Can someone help ...
2
votes
1answer
106 views

Sequences in $\mathcal{L}^p$ and dominated subsequences

For a sequence $\{f_{n}\}$ which converges in $\mathcal{L}^{p}$ space, can we extract a subsequence which is dominated by a function $g \in \mathcal{L}^{p}$? Can anyone help with this? I thought ...
2
votes
0answers
124 views

Generalizing to $\mathbb{R}^n$ a bound for a Lebesgue measure

There's a basic idea in measure theory that if $E$ is any Lebesgue measurable set in $\mathbb{R}$ such that $\lambda(E)>0$, then for any $\epsilon>0$, then there is a nontrivial, finite ...
1
vote
1answer
373 views

prove that f is a characteristic function

$ ( X, \mu) $ is a complete measure space and $E_{n}$ are measurable sets such that $ \mu (E_{n}) < \infty$ for all $n$. Let $ \chi_{E_{n}} $ converge to $f$ in $ L_{1}$. Prove that $f$ is the ...
0
votes
1answer
120 views

Lebesgue measure on boxes with same length sides

I'm considering the following situation. Suppose $\mathcal{A}$ is a semiring of sets in $\mathbb{R}^n$ of the form $(a_1,b_1]\times\cdots\times(a_n,b_n]$. Then there is the unique Lebesgue measure ...
6
votes
2answers
494 views

Omitting the hypotheses of finiteness of the measure in Egorov theorem

I want to prove that if I omit the fact that $\mu (X) < \infty$ in Egorov theorem and place instead that our functions $|f_n| <g$ and $g$ is integrable, we still get the result of Egorov's ...
1
vote
1answer
203 views

If the absolute value of each function in a sequence has a Lebesgue integral that is bounded by a fixed number, will the limit function be integrable?

We have sequence of measurable functions $f_1, f_2, \dots$ such that $f_n \rightarrow f$ a.e. and $f$ measurable. If we know that $\int |f_n|d\mu < B$ for all $n$ ($B$ is fixed and finite). Also ...