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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).

Lebesgue measure is the classical notion of length and area to more complicated sets, and its assigns a measure to subsets of $n$-dimensional Euclidean space. Some examples of Lebesgue any closed interval, any cartesian product of intervals, any Borel set, and any countable set of real numbers (which has Lebesgue measure zero).

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Exercise in prof Tao's "Introduction to measure theory"

I'm struggling with this exercise in Prof Tao's book "Introduction to measure theory". Exercise 1.2.25 (p.43): Define a continuously differentiable curve in $R^d$ to be a set of the form ${\gamma(t):a \leq t \leq b }$, where $[a,b]$ is a closed…
SiXUlm
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Does there exist such a set?

$R$ is a set of real numbers, m is the Lebesgue measure on $R$. Does exist a nowhere dense subset $A\subseteq R $, such that $m(A)=+\infty$? I know that if $A$ is of the first category, there exist such a set. Thanks a lot.
David Chan
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Lebesgue measure theory without Caratheodory's method

Let $U$ be an open subset of $\mathbb{R}$. $U$ is a countable disjoint union of open intervals $I_n$. We denote $m(U) = \sum l(I_n)$, where $l(I_n)$ is the length of $I_n$. Let $A$ be a subset of $\mathbb{R}$. We denote by $\mu^*(A)$ the infimum of…
Makoto Kato
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Limit of Lebesgue measure of measurable function

Let $g: \mathbb{R} \rightarrow \mathbb{R^{+}}$ be any measurable function and for any $\epsilon\geq0$, let $B_{\epsilon}= \{ x\in\mathbb{R}\:\vert\: g(x)>\epsilon\}$. Now show that $$\underset{n\to\infty}{\lim}\lambda(B_{1/n})=\lambda(B_{0})$$ I'm…
Myggen--
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Subset $A \subset [0,8]^2$ with the largest measure satisfying $(A+(3,4) )\cap A=\emptyset$

Find the subset $A \subset [0,8]^2$ with the largest (Lebesgue) measure such that $(A+(3,4) )\cap A=\emptyset$. I tried first rotating the problem so that I'll only have to ensure that $A \cap (A+(5,0))=\emptyset$. But now I'm stuck again.
user195636
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Intersection and Union of two measurable sets

need some help with the following problem in Measure Theory (couldn't find this on the forum) Q. Prove that if A1 and A2 are measurable then $$\lambda(A1 \bigcup A2) + \lambda(A1 \bigcap A2) = \lambda(A1) + \lambda(A2) $$
Parasuram
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$f_n\to f$ in a metric VS $f_n\to f$ in measure.

I have a function $\rho(f,g)$ to be the metric function for any two measurable functions $f,g$. What does it mean by $f_n\to f$ in a metric and $f_n\to f$ in measure, where $(f_n)$ is a sequence of functions.
Shannon
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Considering the parametric space $\Theta = \mathbb{R}^n$, the set $\{ (x_1, \ldots, x_n) \in \mathbb{R^n}: x_1 = x_2 = x_3 \}$ has measure zero?

My question is very direct: considering the parametric space $\Theta = \mathbb{R}^n$, n>3, I want to know if the set $ \{ (x_1, \ldots, x_n) \in \mathbb{R^n}: x_1 = x_2 = x_3\} $ has Lebesgue measure zero in $\Theta$? My intuition say "Yes", for…
Renato Fernandes
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Can this proof be applied to C1-functions

In show that a straight line has a Lebesgue measure of zero, does the proof given applies to every $C^1$-function or does it need some changes for it to be true? More specifically, I want to know if the argument that…
rogueGalileo
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Lebesgue measure on interval

Determine the Lebesgue measure of a set of numbers from the range [0,1] for which there are such decimal increments of $0,1,2,3,\ldots,$ that each of them appears after the decimal point at least once.
Joe
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We know that open intervals are Lebesgue measurable what about close intervals

$M^*(a,b)=b-a$ we know that this fact but how we can prove closed intervals are Lebesgue measurable. I tried to prove by using $\cap ((a-\frac1n),(b+\frac1n))$ But ı totaly stucked :( please help me guys
Manzara
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Why a Collection of disjoint open sets in $\mathbb R^n$ has only countably many nonempty sets?

I am confused by this question. Why can't we just have a disjoint union of open sets of which every set is non empty?
canseeker
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