# 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).

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### If $E$ is Borel, then $\lambda(E)=\sum_{k \in Z} \lambda((E-k) \cap[0,1))$

I'm trying to prove that the measure for a Borel set $E$ can be as $$\lambda(E)=\sum_{k \in Z} \lambda((E-k) \cap[0,1))$$ Where $A-k$ is just a translated set. But I'm having trouble with this. At ...
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### Natural density of sets using Lebesgue measure

Suppose that the sets $A$ and $B$ are specified subsets of positive integers up to $n$. (For instance, $A$ or $B$ could be the set of all even integers less than $n$). Assume also that $A$ and $B$ ...
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### Rudin’s PMA, Theorem 11.20

This is the definition which we need for the theorem: (source) 11.19 $\; \;$ Definition $\; \;$Let $s$ be a real-valued function defined on $X$. If the range of $s$ is finite, we say that $s$ is a ...
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### Intersection of meaning between Lebesque Measure and Natural Density

FYI: the articles linked and their content within are well-known in number theory, so number theory is a tag (correct if need be). $\textbf{Background}$: This article by Maier states that a given ...
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### A space of Lebesgue measure $0$ homeomorph to a space with non-null Lebesgue measure?

Question : Does there exist a space of Lebesgue measure $0$ homeomorph to a space with non-null Lebesgue measure ? My attempt : The problem is I have no idea whether it is true or false. If I were to ...
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### Is the solid of revolution obtained by rotating $E\subset OZY$ Lebesgue-measurable when $E$ is Lebesgue-measurable?

Let $E\subset OZY$ a Lebesgue-measurable set in $\mathbb{R}^2$. How can I prove that the solid of revolution obtained by rotating $E$ around the $Z$ axis is a Lebesgue-measurable set in $\mathbb{R}^3$?...
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### Two elementary questions about integration with respect to measures

I am a self-studying a book about fractal geometry in which some notions of measure theory are reviewed. As one who had zero background in measure theory, I am puzzled about some aspects of what I ...
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### Existence of an element $\xi\in \mathbb R$ such that $m(A\cap (B+\xi))>0$
Let $A$ and $B$ are two positive Lebesgue measurable sets in $\mathbb R,$ that is, $m(A)>0$ and $m(B)>0$, where $m$ denotes the Lebesgue measure in $\mathbb R.$ I want to show that, there exists ...