Mathematics Stack Exchange
Mathematics Stack Exchange

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).

7551 questions
15
votes
2 answers

Why this definition for Lebesgue measurable functions?

A function $f : \mathbb{R} \to \mathbb{R}$ is called Lebesgue-measurable if preimages of Borel-measurable sets are Lebesgue-measurable. I don't understand why we would pick this definition, rather than saying that a function is measurable if…
user735382
  • 153
14
votes
2 answers

Are dense subsets almost nothing or almost everything?

Dense subsets of $[0,1]$ I know have Lebesgue measure $0$ or $1$, but, is there any dense, uniform subset of $[0,1]$ with meausre $1/2$? What I mean with uniform: a subset $A$ of $[0,1]$ is uniform if $m(A\cap[a,b])=(b-a)m(A)$ for $0\le a\le b\le…
ajotatxe
  • 65,084
9
votes
2 answers

The Lebesgue measure of the boundary of a simply connected domain

Is the Lebesgue measure of the boundary of a simply connected domain in $\mathbb{R}^n$ necessarily 0? Acturally, I want to know the sufficient condition to guarantee the measure of the boundary of a domain is $0$.
Lei Zhang
  • 161
7
votes
2 answers

Lebesgue measure of sets of empty interior

There is a proposition to the effect that if the Lebesgue measure of a set is zero, then it is a set of empty interior. Does the converse also hold true, i.e. is the Lebesgue measure of any set of empty interior zero? If not, is there a…
Behrooz
  • 1,000
5
votes
0 answers

Is a Riemann integrable function always measurable?

In many textbooks, it is often written that a Riemann integrable (bounded) function on a compact interval [a,b] is also Lebesgue-integrable, and the two integrals are equal. My question is that, is measurability of the function (w.r.t. the Borel…
Somabha Mukherjee
  • 2,500
  • 1
  • 17
  • 18
5
votes
2 answers

Does a compact set with positive Lebesgue measure mean it containing an open set?

Let $A \subset \mathbb{R}^n$ be a compact set with positive Lebesgue measure on $\mathbb{R}^n$. Can we find an open set $B \subset \mathbb{R}^n$ such that $B \subset A$? PS: I know that if the compactness removed, the answer is no, since $A$ can be…
Ryan
  • 655
5
votes
1 answer

Pointwise multiplication of two measure zero sets

It can be shown that the pointwise sum of two measure zero sets is not necessary of measure zero, take for example the Canter set $C$, we have $C+C=[0,2]$. Now my question is, what about the pointwise multiplication of two measure zero sets in real…
user284331
  • 55,591
4
votes
1 answer

Assume $A \subset \mathbb{R}$ and $m(A) = 0$. Show that $m (A^2) = 0$.

My question is the following: let $m$ denote Lebesgue measure on $\mathbb{R}$. Define $A^2 := \{ x \cdot y \ | \ x, \ y \in A \}$. If $m(A) = 0$, show that $m (A^2) = 0$. My initial observation was to try to write $$A^2 = \bigcup_{a \in A} a \cdot…
Rellek
  • 2,222
4
votes
1 answer

Finding a Lebesgue measurable set that contains a set and they have the same outer measure

Let $A$ be a set in $E=[0,1]\times[0,1]$. Then, there exists a Lebesgue measurable set $A'$ such that $A\subseteq A' \subseteq E$ and $\mu^*(A)=\mu^*(A')$. $\mu^*$ is Lebesgue outer measure. I was searching for a proof for bigger theorem on Lebesgue…
Haley13
  • 83
3
votes
1 answer

Convergence of a sequence of non-negative values

Let $\{F_k\}_k\subset[0,1]$ be a collection of sets such that $m(F_k)\geq \delta$ for some $\delta>0$ for all $k$. Let $\{a_k\}_k$ be a non-negative sequence such that $\sum_{k=1}^{\infty}a_k\chi_{F_k}(x)<\infty$ for almost all $x$. Show that…
aqwer
  • 153
3
votes
1 answer

Lebesgue measure of $\mathbb{Q}\cap [0,1]$

For $E\subseteq\mathbb{R}^n$ I was given this definition of Lebesgue measure: $m(E)=\sup \{m(K) \mid K\subseteq E \text{ compact}\} = \inf \{m(A) \mid A\supseteq E \text{ open}\}$ if they are equal, where: $m(K)=\inf\{m(P) \mid P\supseteq K \text{…
rod
  • 769
3
votes
1 answer

Lebesgue Measure of $S_k$

let $S_k$ is the set of elements of $(0,1)$ whose kth position is prime. Then what's is the lebesgue measure of $S_k$ I don't know how to approach the question... Any help leading to answer is deeply appreciable Thanks
user485546
3
votes
0 answers

Lebesgue measure beyond Lebesgue measurable sets

Is it possible to extend the Lebesgue measure as a translation-invariant measure from the Borel sigma-algebra to a sigma-algebra not contained in the Lebesgue sigma-algebra?
user60121
  • 123
3
votes
1 answer

Lebesgue measurable subsets of [0,1]

If $A$ is a Lebesgue measurable subset of $[0,1]$ such that $\lambda(A)>0$. Show that there are Lebesgue measurable subsets $A_{s},0\leq s \leq \lambda (A)$, so that (a) $A_{r}\subseteq A_{s}$ if $r\leq s$, (b) $\lambda (A_{s})=s$ for any $0 \leq s…
LanaDR
  • 503
3
votes
1 answer

Measure Lebesgue-Stieltjes of single point in $\mathbb{R}$

If $f$ is a nondecreasing function defined on $\mathbb{R}$, then the “length” of a half-open interval $(a, b]$, denoted by $\alpha_f((a;b])$ can be defined by $\alpha_f((a;b]) = f(b) - f(a)$ The Lebesgue-Stieltjes outer measure of an arbitrary…
user402085
  • 492
1
2 3
…
10 11