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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 Borel measurable function composed with Lebesgue measurable function a Lebesgue measurable function?

Let $g$ be a Borel measurable function and $f$ be a Lebesgue measurable function. Then, is $g(f(x))$ a Lebesgue measurable function?
Analysis
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Lebesgue measure: Show $m(\cap_{k=1}^{2014} E_k)>0$...

Here's the full problem: Let the collection of measurable sets $\{E_k\}_{k=1}^{2014} \subset [0,1]$ be s.t $\sum_{k=1}^{2014} m(E_k) >2013$, where $m$ denotes Lebesgue measure. Show $$m(\cap_{k=1}^{2014} E_k)>0$$ Here is what I've been…
user146925
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Question on Lebesgue outer measure

The question is as follows : Show that for any set $A$ and $\epsilon > 0$, there is an open set $O$ containing $A$ and such that $m^*(O) \le m^*(A)+ \epsilon$. Solution given in the book is as follows : Choose a sequence of intervals $I_n$ such that…
johny
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meas-square integrable function that is not integrable

F the Borel sets, and u Lebesgue measure Show that there exists a mean-square integrable function on X that is not integrable if X=R
user1713792
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Lebesgue measure and area of a set

So the question is something like this: If a subset of $\mathbb{R}^2$ has a Lebesgue measure 0 and HAS an area. Does that area have to be 0? Prove or disprove. I know of counterexamples that don't even have an area defined but I figured that since…
Luka Horvat
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property of Lebesgue measure

Suppose that $E \in \mathcal{M}$, show that for each $\epsilon > 0$ there is a closed set $F$ such that $F \subset E $ and $\lambda(E \setminus F) < \epsilon$, where $\mathcal{M}$ is the collection of Lebesgue measurable sets and $\lambda$ the…
ywx
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Lebesgue measure unbounded sets

I have a question which asks if the following is true: if $A,B\subseteq \mathbb{R}$ are closed, disjoint sets that are unbounded, then $m^*(A\cup B)=m^*(A)+m^*(B)$. I think it is true due to the fact that $A$,$B$ are closed which implies they are…
Eduardo
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Lebesgue measure of a bounded set

This seems trivial but how would you prove that an open and bounded set, G, has finite lebesgue measure.
Anonymous
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Clarification regarding Fubini's theorem

In Stein's textbook Chapter 2, page 75-76. We read: "...even the assumption that $f$ is measurable on $\Bbb{R}^d$, it is not necessarily true that the slice $f^y$ is measurable on $\Bbb{R}^d$ for each $y$; nor does the corresponding assertion…
ISO
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Why do we need to take infinitely many cuboids in the definition of the Lebesgue outer measure?

I fear this question has already been asked but I couldn't find anything with the search tool. Our definition of the Lebesgue outer measure on $\mathbb{R}^n$ is $$ \nu (A)= \inf \{ \sum_{I=1}^\infty Vol(Q_i) | Q_i \in \mathcal{Q}_n ,A\subset…
The Lion King
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Approximation of Lebesgue measurable set by compact with entry interior

Let $E \subset \mathbb{R}$ Lebesgue measurable, finite and let $\epsilon>0$. Show that there exists compact $K\subset E$ s.t. $m(E-K)<\epsilon$, and $K^0= \emptyset $. For K with not necessary entry interior I know it is a classic result, but with…
Studentmathever
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Show that $\phi$ is continuous on $E$.

Let $f:X \to C$ be measurable, $||f||_{\infty} > 0$, and let $\phi(p)= \int_X|f|^pd\mu$, and let $$E=\{p \in (0,\infty): \phi(p) < \infty\}$$ I would like to show that $\phi$ is continuous on $E$. I am really stuck on this. Any helps would be…
akai
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How to prove this problem about lebesgue measure?

Let E⊆R and have outer measure.Show that there is an Fσ set F and a Gδ set G Such that F⊆E⊆G and m*(F)＝m*(E)=m*（G）. I can prove there is G Such that E⊆G and m*(E)=m*(G) by definition of outer measure to get ∪Iₙ and E⊆∪Iₙ, Such that m*(∪Iₙ)…
user793548
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Upper bound Lebesgue measure of "square" of set on closed interval

This is a follow up to this question: Upper bound "square" of Lebesgue measure of set Let $\lambda$ be the Lebesgue measure, $A$ a set with $\lambda(A) < \varepsilon$ and $M > 0$. Consider the set $A^2 = \{a \cdot a \ | \ a \in A\}$. Can we produce…
Lundborg
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Every real continuous function defined on a Lebesgue measurable set is Lebesgue measurable

For a given Lebesgue measurable set $E$ and a continuous function $f\colon E \longrightarrow \overline{\mathbb{R}},\ $ I am asked to prove that $f$ is Lebesgue measurable. My Proof: If $f$ is Lebesgue measurable, the set $A = \{ x \in E \mid…
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