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Questions tagged [lebesgue-measure]

For questions about the Lebesgue measure, a measure defined on the Borel or Lebesgue subsets of the real line or $\mathbb R^d$ for some integer $d$. Use it with (tag: measure-theory) tag and (if necessary) with (tag:lebesgue-integral).

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Translates of a set of positive Lebesgue measure cover $\mathbb{R}$?

Let $E$ be a set of positive Lebesgue measure in $\mathbb{R}$. Does some countable union of translates of $E$ cover $\mathbb{R}$? My intuition is that $\mathbb{R}$ can be covered with countable ...
CeyhunElmacioglu's user avatar
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How to find a measurable, Lebesgue invariant bijection between the interval and the 3-dimensional sphere

From a book I was reading I have the following statement: "It is a general fact of measure theory that there is a bijection $f : [0, 1) → \mathbb{S}^2$ such that both $f$ and $f^{-1}$ take ...
Riel Blakcori's user avatar
1 vote
0 answers
63 views

Rudin's RCA 2.24 theorem : Lusin's theorem

As precised in the title of my question, the context is the book of Walter Rudin : Real and Complex analysis. And especially the proof of theorem 2.24 (Lusin's theorem) which I put below. I have a ...
Laurent Garnier's user avatar
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How to prove a Fourier transform inequality for a function with bounded support in mixed variables?

I'm working on a problem involving the Fourier transform and have encountered an inequality that I am unsure how to prove. I would greatly appreciate any help or guidance you can provide. Let $\gamma\...
logarithm's user avatar
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1 answer
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The set of the points where a measurable function is continuous is measurable

Let $f:\mathbb{R}^n \rightarrow \mathbb{R}$ be a Lebesgue-measurable function and let $C = \{ x \in \mathbb{R}^n | f \text{ is continuous in } x \}$. Show that $C$ is Lebesgue-measurable. I found a ...
F13's user avatar
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Is there a model of ZF not C where not every set of reals is Lebesgue measurable? [duplicate]

I know that there is a model of ZF set theory plus the negation of the axiom of choice where every set of reals is Lebesgue measurable. But is there also a model where not every set of reals is ...
user107952's user avatar
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4 votes
1 answer
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Approximation of a class of measurable functions by simple functions with "compact domain"

It is well known that if $f:\mathbb{R}\rightarrow \mathbb{R}$ is a measurable function and $f \ge 0$ then there exist a sequence of simple non-negative measurable functions {$ S_n $} such that $S_n\...
iki's user avatar
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1 vote
2 answers
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Find a sequence of $Lˆ1$-summable functions with $|f_n(x)| \leq 1$ and some other properties

I am studying measure theory and summability/integrability in particular. One exercise is to find a sequence of functions $\{f_n\}_n$ on $(0,\infty)$ satisfying the following criteria: i) $\{f_n\}_n$ ...
arridadiyaat's user avatar
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0 answers
26 views

Invariance under measure preserving functions

Let $W: [0,1]^k \to \mathbb{R}$ be a Borel-measurable function such that $W$ is invariant under any measure-preserving function $\varphi: [0,1] \to [0,1]$, i.e. for $W^\varphi(x_1, \dots, x_k) := W(\...
DDD's user avatar
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find measure of set $E=\cap_{n=1}^{\infty}\cup_{k=n}^{\infty} E_k.$ where $E_k$ has measure $\frac{1}{k^2}$.

For $𝑘 ∈ ℕ$ , let $𝐸_𝑘$ be a measurable subset of $[0,1]$ with Lebesgue measure $\frac{1}{k^2}$ Define $$E=\cap_{n=1}^{\infty}\cup_{k=n}^{\infty} E_k.$$ $$F=\cup_{n=1}^{\infty}\cap_{k=n}^{\infty} ...
Ricci Ten's user avatar
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Lebesgue outer measure in $\mathbb{R}^2$ in terms of a grid of $h$-squares

For a set $D\subseteq\mathbb{R}^2$, the Lebesgue outer measure of $D$ is defined by $$\lambda^\ast(D)=\inf\bigg\{\sum_i\lambda(I_i)\mid D\subseteq\bigcup_iI_i\bigg\},$$ where $\{I_i\}$ is a sequence ...
ashpool's user avatar
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Definite integral of Modified Bessel function, exponential and trigonometric functions

I am trying to solve the following integral; $$ \int_{0}^{\frac{\pi}{2}} e^{\gamma \cos\theta} I_{1}(\epsilon\sin\theta)d\theta,$$ where $\gamma\in\mathbb{R},\epsilon\in\mathbb{R}^{+},$ and $I_{1}$ is ...
Nelly Clark's user avatar
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How to compare size of $\sigma$-algebras on $\mathbb{R}$, or more generally sets of sets of $\mathbb{R}$?

I understand that this is a peculiar question that might not be very enlightening, but I am trying to compare how much bigger the set of Lebesgue measurable sets are compared to the set of Borel sets, ...
Sai Nallani's user avatar
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What's the definition of a line integral on a possibly disconnected curve?

I'm trying to understand this paper, and I see this integral (page 2): $$ \int_{B\ \cap\ \mathcal{C}} (1 - y)dy, $$ where $\mathcal{C}$ is the curve given by $x = ye^{1-y}$ and $B$ can be any ...
Polygon's user avatar
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measurability of maximal sets of pseudo-injectivity of continuous function

For a given measurable set $\Omega\subseteq {\bf R}$ such that $\lambda(\Omega)>0$ and $n\in {\bf N}$, we define (maximal) sets $U_n\subseteq\Omega$ as follows: $$ U_n:=\{\sigma\in\Omega:\quad {\rm ...
Andrija's user avatar
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1 answer
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Rudin $8.18$ theorem.

There are necessary definitions and theorem for the proof: here is the theorem: If $1\lt p \lt \infty$ and $f \in L^{p}(R^k)$ then $Mf \in L^{p}(R^k)$ there is its proof: Since $Mf = M(|f|)$ we may ...
JohnNash's user avatar
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when does $m(A/E)=m(A)-m(E)$?

I have to show that $E \in M \iff m(A/E)<\epsilon$ $\forall$ A open set $\supset$ E for $\leftarrow$ is ok the other direction I know that $m(E)=inf_{A \supset E , \text{A open} }m(A)$ so $\exists ...
Dsrksidemath's user avatar
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3 answers
64 views

Function $f:[0,1] \rightarrow \Bbb R$ such that $\forall c \in \Bbb R: f^{-1}(c)$ Lebesgue Measurable, but $f$ not Lebesgue Measurable.

I have come up with a counterexample to the following statement: Let $f:[0,1] \rightarrow \Bbb R$ and suppose that for every $c\in \Bbb R$, the set $\{x \in [0,1]\ |\ f(x)=c\}$ is $\mathcal L^1$-...
clorx's user avatar
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Is $\ln(ax^2+y^2)\in L^2_{\operatorname{loc}}(\mathbb{R}^2)$?

Let $a>0$ and define $$f(x,y)=\ln(ax^2+y^2)$$ for each $(x,y)\in\mathbb{R}^2\backslash\{(0,0)\}$. Is $f$ in $L^2_{\operatorname{loc}}(\mathbb{R}^2,\mathcal{L}^2)$, where $\mathcal{L}^2$ is the two-...
Twnk's user avatar
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1 vote
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111 views

Mollification and limit

Consider the $d$-dimensional torus $\mathbb{T}^d.$ Let $\phi \in C_c^\infty(\mathbb{R}^d,\mathbb{R})$ such that $\phi(0)=1$ and $\phi_\epsilon(x):=\phi(\epsilon x),\rho\in C^\infty(\mathbb{T}^d).$ ...
mathex's user avatar
  • 616
3 votes
0 answers
47 views

Is the Evaluation Map on Bounded Borel Measurable Functions Borel Measurable?

I am working with a set $I$, defined as the closed interval $[0,1]$, and a set $X$, which consists of all bounded Borel measurable functions defined on $[0,1]$. The uniform metric on $X$ is defined by:...
f yz's user avatar
  • 51
2 votes
1 answer
154 views

Continuous function which satisfies the Luzin N property, but which does not satisfy the Banach S property

My question is: can we find the function $g\in{\rm C}([a,b])$ which satisfies the Luzin N property on $[a,b]$, but which does not satisfy the Banach S property on [a,b]? Here $[a,b]$ is a compact non-...
Andrija's user avatar
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0 answers
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Lebesgue measure generating set

In my cource of measure theory, Lebesgue measure were built starting with measure on semiring $S\subset 2^X$, after we defined outer measure for each $A\in2^X$ by $\inf\{ \sum{m[B] : A\subset \cup B , ...
nagvalhm's user avatar
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2 answers
101 views

if $f_n \uparrow f$ prove $\mu(f_n\geq t)\rightarrow \mu(f\geq t)$

I think I have to use montone convergence theorem and I have followed this line the sets $f_n\geq t$ are increasing $\lim\mu(f_n\geq t)=\mu(\cup(f_n\geq t)=?\mu(f\geq t)$ if $x\in \cup(f_n\geq t) \...
Dsrksidemath's user avatar
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0 answers
62 views

Proof that the volume function is $\sigma$-additive

A $\textbf{box}$ $Q\subset \mathbb{R}^n$ is defined as $Q:= (a_1,b_1) \times \ldots \times (a_n,b_n)$ and the same with closed or half opened intervals where $a_i < b_i \in \mathbb{R}$ and in case ...
Roger Crook's user avatar
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0 answers
25 views

Standard measure and Standard topology on the set $[0,\infty]$

I'm currently trying to follow along with some lecture notes on probability and measure theory, but I need some help understanding some definitions Let $f: E \rightarrow G$ be a measurable function ...
Ogglie Ostrich's user avatar
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0 answers
42 views

Lebesgue Measure - Disk

Probably a very easy question, but still want to check if I understood correctly. Let $F$ be the unit flat disk in the $xy$-plane in $\mathbb{R}^3$. $$ F=\left\{(x, y, z) \in \mathbb{R}^3 \mid x^2+y^2 ...
Mikeys00's user avatar
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0 answers
32 views

What is the theory of the measure of a line in a plane

I have been taught how to measure the length of a line in a plane, in typical situations encountered in physics; for instance, by integrating $dl = \sqrt{dx^2+dy^2}$ along the line, parametrized in ...
Twizzle's user avatar
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1 vote
2 answers
49 views

$L^\infty(\Omega)$ is dense in $L^{p,\infty}(\Omega)$ if $\Omega$ is compact

Given a compact set $\Omega\subset \mathbb{R}^N$, I am wondering if $L^\infty(\Omega)$ is dense in the weak $L^p$ space $L^{p,\infty}(\Omega)$ with $1< p<\infty$ (see here the definition). I ...
mejopa's user avatar
  • 433
6 votes
1 answer
479 views

Let $f\colon (a,b)\to \mathbb{R}$ be nondecreasing and continuous. If $E=\{x\in (a,b)\mid f'(x)\text{ exists and } f'(x)=0\}$, then $\lambda(f(E))=0$

I need help to understand the proof below of the following theorem. Let $f\colon (a,b)\to \mathbb{R}$ be an arbitrary function. If $E=\{x\in (a,b)\mid f'(x)\text{ exists and } f'(x)=0\}$, then $$\...
NatMath's user avatar
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-2 votes
0 answers
31 views

Fourier Transform on $R^2\backslash (0,0)$ [closed]

I'm a theoretical physicist and in my research I have functions $\psi(p)$ where $p \in \mathbb{R}^2 \backslash \{(0,0)\}$ I obtained them through Mackey theory of induced representations and they host ...
LuVa's user avatar
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0 votes
0 answers
25 views

Integral of a function w.r.t. Lebesgue measure

In a certain exercise I have been asked to prove that $\int_{1}^{\infty}\frac{1}{x}dm = \infty$, where $m$ is the Lebesgue measure. It is the first time that I study measure theory, so I'm really lost....
Emmy N.'s user avatar
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9 votes
1 answer
169 views

Let a real measurable function $f$ map every open set to the whole real line. Is there always a restriction to a set of measure zero doing the same?

Given $f:\mathbb{R}\rightarrow\mathbb{R}$ mapping each non-empty open set to the whole real line, is there always a set $A$ of measure zero such that the function $g:\mathbb{R}\rightarrow\mathbb{R}$ ...
Adam's user avatar
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2 votes
1 answer
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Is there a function $f:\mathbb{R}\rightarrow\mathbb{R}$ that maps every subset of cardinality $|\mathbb{R}|$ to the whole real line?

Functions like Conway's base-13 function map each open set to the whole real line. Functions like this one I think map every set of positive Lebesgue measure to the real line but I don't know how to ...
Adam's user avatar
  • 335
2 votes
1 answer
88 views

What is the measure of the set of values $S=\{0.x_1x_2... \in (0,1)$ (in binary) for which $\sum_{n \geq 1}\frac{(-1)^{x_n}}{n}$ converges$\}$?

I think the measure is zero but I'm not sure. I'm pretty sure the set is measurable because its construction does not require Axiom of Choice. I don't know where to begin. Edit: It was shown to be 1 ...
Adam's user avatar
  • 335
0 votes
0 answers
39 views

Is a real line an elementary set?

Now I'm studying Measure party from Rudin's PMA. It says that, if a set $A$ can be written as a union of finite number of intervals, then $A$ is an elementary set. (Now, I will denote the colletion of ...
Irohas's user avatar
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0 votes
1 answer
33 views

Ordinal Event Ordering $P(E)\geq P(E')\iff Q(E)\geq Q(E')$, so we can conclude something like $P$ and $Q$ are the same measure?

Let $(S,\Sigma, P)$ be a usual probability space. $S=[0,1]$ and $P$ is the usual Lebesgue measure. $E\in \Sigma$ is an event. Given $P(E)\geq P(E')\iff Q(E)\geq Q(E')$, Is it possible that $P$ and $Q$ ...
dodo's user avatar
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0 votes
1 answer
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$\mu[X\geq x]\geq \mu[Y\geq x] \iff \nu[X\geq x]\geq \nu[Y\geq x] $ implies that the two measures $\mu,\nu$ are the same measure?

$X:[0,1]\to\mathbb R$. $X,Y$ are finite-valued random variables. There are two different measures, $\mu,\nu$, on $S=[0,1]$. $\mu$ is the Lebesgue measure. First order stochastic dominance FOSD is ...
dodo's user avatar
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0 votes
0 answers
20 views

How can I prove Lebesgue's density theorem?

I'm trying to find the proof in every Euclidean real space $R^d$, and what I found is this ($D(E)$ is the set of points which makes the density of $E$ to $1$.): $$$$ We will prove that $m(E \setminus ...
Strassss's user avatar
0 votes
1 answer
62 views

Poincaré inequality for generic measure?

Let $\Omega$ a smooth bounded domain, and let $\mu$ an absolute measure with respect the Lebesgue measure. I need to prove the following Poincaré inequality (if it is true!) $$ \int_\Omega f^2 d\mu \...
Emma Notes's user avatar
1 vote
0 answers
22 views

Lebesgue measure of a set of limit points of a set?

Is there any subset of $\mathbb R^d$ that the set of its limit points has Lebesgue measure greater than zero?
InfSuplife's user avatar
2 votes
1 answer
89 views

Rudin’s RCA Theorem $7.26$ part $1$.

These are definitions which we use in the proof of the theorem: This is lemma $7.25$ which is used in the proof: There is the theorem: Suppose that $(i)$ $X \subset V \subset R^{k}$, $V$ is open, $T ...
JohnNash's user avatar
  • 1,020
1 vote
1 answer
45 views

Complemented subspace of set o functions whose integral is zero

If we consider the space $L^1([0,1])$, and its subspace: $$ Y = \left \{ f \in L^1([0,1]) : \int_{[0,1]} f d \mu = 0 \right \}$$ What would be the complemented subspace of Y, i.e. set Z such that: $$ ...
AlaskaYoung's user avatar
0 votes
1 answer
52 views

Is a Borel measurable function always Lebesgue measurable? [duplicate]

Given the following definition of a measurable function: Let $(X, \mathfrak{M})$ and $(X, \mathfrak{N})$ be two measure spaces. A function $f:X \longrightarrow Y$ is $(\mathfrak{M} , \mathfrak{N)}$-...
Mulstato's user avatar
0 votes
1 answer
55 views

The measure of a set of continued fractions. [duplicate]

What would be the measure of the set of continued fractions which only contain 1 or 2 in the denominator, including infinite fractions? If I understand measure correctly, the set of continued ...
nnabahi's user avatar
  • 83
2 votes
1 answer
60 views

If $\int_0^1 f(x) \, dx = I$, then $m\left\{x: f(x) > \frac{I}{2}\right\} \geq \frac{I}{2}$

Consider a measurable function $f:[0,1] \to [0,1]$ such that $\int_0^1 f(x) \, dx = I$. Prove that $$m\left\{x: f(x) > \frac{I}{2}\right\} \geq \frac{I}{2} \tag{$\star$}$$ I'm not sure how to go ...
Grigor Hakobyan's user avatar
0 votes
0 answers
32 views

Computing the CDF associated to a random variable, which is neither discrete nor continuous, by using the pushforward measure and transfer theorem.

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and $Z$ a real-valued random variable on this space to $(\mathbb{R},\mathcal{B}(\mathbb{R}))$. Let $Z\sim \frac{1}{3}\delta_{\frac{1}{2}} + ...
Jürgen Sukumaran's user avatar
1 vote
0 answers
31 views

Collections of distinct subsets of an interval (existence)

Question: does there exist an uncountable family of distinct subsets of $[-1, 1]$, denoted by $(U_j)_{j \in [-1, 1]}$, with the following properties? Throughout, $\mu$ is just the Lebesgue measure. ...
Stepan Plyushkin's user avatar
0 votes
1 answer
74 views

The area of the area bounded by curve: $ \{ (x,y) \in \mathbb{R}^2 | x > 0, y < 0, x^4 -xy^2 - y^3 = 0 \} \cup { (0,0) }$, wrong parametrization?

Show that the set: $$ \{ (x,y) \in \mathbb{R}^2 | x > 0, y < 0, x^4 -xy^2 - y^3 = 0 \} \cup { (0,0) }$$ is a closed curve. Calculate the area of the area bounded by this curve. Hint: substitute: ...
thefool's user avatar
  • 1,064
1 vote
1 answer
30 views

Given $f \in L^p_{\text{loc}}(\Omega) \setminus L^\infty(\Omega)$, does it follow that $A \cap S(f,K) \neq \emptyset$ for all $K > 0$?

Context. Throughout this post I will be dealing with the Lebesgue measure over $\mathbb R^n$. Moreover, I denote the measure of a measurable set $E \subset \mathbb R^n$ by $|E|$ and $\Omega \subset \...
xyz's user avatar
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