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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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Is it true that the Lebesgue measure of the boundary of any set is zero?

Is it true that the Lebesgue measure of the boundary of any set is zero? If no, then what are the counterexample and what are the conditions under which the above statement is true? Like for example, this book says that dimension of the boundary of…
user297008
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Construct a function that is not Lebesgue-integrable but it is Riemann-interagrable

I must construct a function $f: [0,\infty) \to \mathbb{R}$, but not Lebesgue-Integrable, but $f$ in $[0,c)$ is Riemann-integrable and $$\lim_{c \to \infty} \int_{0}^{c} f(x)dx$$ exists. What I am thinking is: $f(x)=1/x$ on $\mathbb{R}$ and $f(x)=c,…
Melina
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Does $f$ measurable $\iff$ $|f|$ measurable?

Does $f$ measurable $\iff$ $|f|$ measurable ? $\implies $ is clearly true since $$|f|^{-1}((-\infty ,\alpha))=f^{-1}(]-\alpha,\alpha[)$$ and thus $|f|^{-1}$ is measurable. But for the reciprocal I have doubt. I would says yes if $f$ measurable…
Rick
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Regularity of Lebesgue outer measure

The terminology in this area is somewhat confusing, my question is how to prove: Given $E \subseteq \mathbb{R}$, there exists a Lebesgue measurable set $A$ such that $E \subseteq A$ and $\lambda^*(E) = \lambda^*(A)$. Some authors call such $A$ a…
SAP
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Lebesgue measure of a subspace of lower dimension is 0

I'm currently reading Rudin's book Real and Complex analysis. In page 52 he says To prove (e) let $T:R^k\to R^k$ be linear. If the range of T is a subspace Y of lower dimension then $m(Y)=0$. I don't quite get that part. I'm guessing it could be…
Zero
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Exercise 3.D. in Robert G. Bartle's book

Let us consider the problem 3.D. in the book $\textbf{The Elements of Integration and Lebesgue Measure}$ of Robert G. Bartle Let $X=\mathbb{N}$ and $\mathcal{A}$ be the $\sigma-$algebra of all subsets of $\mathbb{N}$. If $(a_n)$ is a sequence of…
impartialmale
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Prove $f^{-1}(B)\in\mathcal{A}$ where Borel set $B$, $\sigma$-algebra $\mathcal{A}$, $\mathcal{A}$-measurable function $f$.

I am trying to solve a homework problem in "Lebesgue integration" course. And I found the theorem seems to be useful to my problem which is : Let $\mathcal{A}$ be a $\sigma$-algebra of $\mathbb{R}$, $B$ be a Borel set of $\mathbb{R}$, and…
Analysis
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Lebesgue outer measure of a finite set

I was trying to work out the Lebesgue outer measure of a finite set. I took a simple example, considered the finite set $\{a,b\}$. Then, I wrote $\{a,b\}$ as $\{a\} \cup \{b\}$. Since the Lebesgue outer measure of singleton sets is zero, therefore,…
johny
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When the upper sum $U[f;P]$ equals the lower sum $L[f;P]$ for a measurable partition $P$

I know if there exists a measurable partition $P$ of $[a,b]$, $f$ is Lebesgue integrable on $[a,b]$ when $$ \inf_P U[f;P] = \sup_P L[f;P]$$ where the infemum and supremum are over all measurable partitions. However, what can I say the function f if …
user140744
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Limits of the Lebesgue measure

Let $(\mathbb{R}^d, \mathfrak{M}, m)$ be Lebesgue measure space and $A \subset \mathfrak{M}$ with $m(A) < \infty$. Suppose $f : \mathbb{R}^d \rightarrow \mathbb{R}$ as $f(x)=m(A \cap(x+A))$ with $x+A=\{x+a: a \in A\}$ I want to show $\lim _{|x|…
alryosha
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Upper bound "square" of Lebesgue measure of set

Let $\lambda$ be the Lebesgue measure and $A$ a set with $\lambda(A) < \varepsilon$. Consider the set $A^2 = \{a \cdot a \ | \ a \in A\}$. Can we produce an upper bound on $\lambda(A^2)$ in terms of $\varepsilon$?
Lundborg
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Covering all rationals in (0,1) by open intervals of total lenght epsilon. Is there any real in (0,1) not in the covering?

It seems we can cover each rational in (0,1) by a set of open intervals of decreasing lenghts whose sum is arbitrarily small. So it looks like there are a lot of holes. But on the other side, rationals are dense on (0,1)... Does it mean the…
Eduard
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Proving $E$ is measurable iff $E^C$ measurable... using the alternative definition

I'm interested in potentially using the following definition to develop the basic theory of Lebesgue measurable sets, but I'm running into a considerable roadblock. First, some context Definition: A set $A \subseteq \mathbb{R}^d$ is Lebesgue…
Thomas Winckelman
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Lebesgue measure of a subspace of lower dimension

I'm aware that this question has been asked before by Zero. However, there is a step in the answers provided by others that I've yet to understand. Let $T:\mathbb{R^n} \to \mathbb{R^n}$ be linear. If the range of $T$ is a subspace $Y$ of lower…
Jia Cheng Sun
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Intuition behind the definition of Lebesgue measurable function

In Real Analysis written by Royden, the definition of measurable function is as follows. An extended real-valued function f defined on E is said to be Lebesgue measurable, or simply measurable, provided its domain E is measurable and it…
alryosha
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