# 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 ...
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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 ...
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### 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 ...
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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) $\{f_n\}_n$ ...
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### 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 $\mathcal L^1$-...
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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-...
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### 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 $\phi_\epsilon(x):=\phi(\epsilon x),\rho\in C^\infty(\mathbb{T}^d).$ ...
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### 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:...
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### 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-...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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....
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### 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 $g:\mathbb{R}\rightarrow\mathbb{R}$ ...
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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 ...
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### 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 ...
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### 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 ...
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### 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 $Q$ ...
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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 ...
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### 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. ...
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### 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: ...
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### 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 \...