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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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For a measure space prove if $f$ is measurable and bounded then there exists simple functions that converge to it?

Let $(X,\Sigma)$ be a measure space and $f:X\rightarrow \mathbb{R}$ be measurable and bounded. Then there exists a sequence of simple measurable functions $f_n$ such that $f_n\rightarrow f$ uniformly, ie: $$\forall\epsilon>0 \mbox{ } \exists N…
Daniel
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Rotation send measure zero set to measure zero

$m_2$ is the Lebesgue measure on $\mathbb R^2$. Suppose U is a zero set. $T:\mathbb R^2 \to \mathbb R^2$ is a rotation. Show $T(U)$ is a zero set. What I tried is to show the outer measure defined by open rectangle cover is equivalent to the outer…
user136592
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Extention of Lebesgue measure

For $n \in \Bbb N$, can we extend the Lebesgue measure to some measure(or complete measure) on $\Bbb R^n$? Can we extent the Lebesgue measure to some measure on the power set of $\Bbb R^n$? (The answer is negative for $n=3$, by using Banach- Tarski…
Arman
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Approximation of Measurable Set

Question 18 of Section 2.5 (page 43) in this link : http://math.harvard.edu/~ctm/home/text/books/royden-fitzpatrick/royden-fitzpatrick.pdf claims that if a set $ E $ has finite outer measure, then, there is a set $ F \in F_{ \delta} $ such…
Chern
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Show that $m(E_1 \cup E_2) + m(E_1\cap E_2)=m(E_1)+m(E_2)$...

Here is the full problem: Suppose $E_1,E_2 \subseteq \mathbb{R}$ are measurable sets. Show that $$m(E_1 \cup E_2) + m(E_1\cap E_2)=m(E_1)+m(E_2)$$ My attempt has been as such: Suppose $E_1, E_2$ are measurable. Now, recall that $$E_1\cup E_2 =…
Savage Henry
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How to show that $m(E_1 \cap E_2)+m(E_1 \cup E_2)=m(E_1)+m(E_2)$?

Let $E_1$ and $E_2$ measurable sets. Show that $m(E_1 \cap E_2)+m(E_1 \cup E_2)=m(E_1)+m(E_2)$ if $E_1$ is measurable, for any set A, $m^*(A)=m^*(A\cap E_1)+m^*(A \cap E_1^c)$ how can I start? (sorry my jvs isnt working I cant add correct tag)
lyme
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Measure of a strange set on $S^{p-1}$

Let $S^{p-1}$ be the unit sphere in $\mathbb R^p$, $s \in \mathbb N$ and \begin{align*}A_s :&= \{ t \in S^{p-1} \text{ such that } t_1 > ... > t_p > 0 \text{ and } t_1 + ... + t_s \geq t_{s+1} + ... + t_{p} \} \\ & = \{ t \in \mathbb R^p \text{…
user130949
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Example of how the measure of a measurable set can be bounded

In one book, the Lebesgue measure is said to possess the following properties: 1) the measure of any measurable set can be approximated from above by open sets; that is, for any measurable M $\mu(M) = inf \{ \mu(O): M \subset O, O$ is open} 2) the…
nayriz
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$\sigma$-additivity of Lebesgue measure.

Can one show that Lebesgue measure is $\sigma$-additive on using only its definition which is $\lambda^n ([a,b))= \prod_{i=1}^n(b_i-a_i)$ and the fact that the set of semi-open boxes form a semi-ring? I saw a proof in a book which relies way too…
user63697
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Is the union of any set of Lebesgue null test sets a measurable set?

I know that the union of countably null sets must be measurable and have measure 0, but what happens when the number of sets is uncountable? For example, the union measure of all single point sets on [0,1] is 1, so the measure changes after taking…
xiaomao
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Lebesgue integrals measure of set

If a
Calculas
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Proof of Vitali Theorem in Jech's The Axiom of Choice uses a questionable statement about Measure

Thomas J. Jech, on the second page of his The Axiom of Choice (1973) proves the existence of a nonmeasurable set of real numbers thus: Let $\mu(X)$ denote the Lebesgue measure of a set $X$ of real numbers. We know that $\mu$ is countably additive…
Prime Mover
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Characterization of Lebesgue outer measure

In L. Simon's book "Introduction to Geometric Measure Theory", one can find the following characterization of the Lebesgue outer measure (in $\mathbb{R}^n$) on page 11: It is the unique outer measure $\mathcal{L}^n$ such that $\mathcal{L}^n\left(…
vizietto
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When is a subset Lebesgue measurable?

I have a task where i should check if subsets are Lebesgue measurable and if the Lebesgue measure is finite. One of the subsets is: $$B:=\bigcup\limits_{k=1}^{\infty} \left(\frac{k-1}{k+1},\frac{k}{k+2}\right]\times…
Peirot
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Is measure of $\bar{E}$ always equal to zero, where $E$ has measure $0$ and nowhere dense in $\mathbb{R}$

let E is subset of $\mathbb{R}$ ,where E is nowhere dense and outer measure of $E=0$. Then outer measure of $\bar{E}=0$? I think is there exist a subset of $\mathbb{R}$ that is nowhere dense set and has measure zero whose closure is equal to…
Tony
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