# Questions tagged [measure-theory]

Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.

1,606 questions
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### what is the associated Borel set of a Borel measurable function on the extended real line?

This question comes from Theorem 19.B in page 81 of Halmos' "Measure Theory", as the image below shows. In this theorem, we are given a function $\phi$ which is a Borel measurable function on the ...
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### Reference for the Gelfand-Neumark theorem for commutative von Neumann algebras

The Gelfand-Neumark theorem for commutative von Neumann algebras states that the following three categories are equivalent: (1) The opposite category of the category of commutative von Neumann ...
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### Borel vs measure for all Borel measures

Let X be locally compact and Hausdorff, and let $f:X\rightarrow\mathbb R$ be a function. Suppose that for all finite regular (positive) Borel measures $\mu$, we know that $f$ is $\mu$-measurable. ...
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### Percolation Model and Complex Probabilities

Let $d>0$ be an integer and consider the first neighbors independent bond percolation model in $\mathbb Z^d$, where each edge is open with probability $p\in[0,1]$. I would like to know, if can we ...
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### disjointlize an arbitrary sequence in a ring?

In a ring R (nonempty class of sets closed under difference and finite union), any sequence (here means a function on natural numbers $\mathbb N$) {$E_i$} in R can be disjointlized to a disjoint ...
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### terminology about ring/algebra in abstract algebra and measure theory

Both in abstract algebra and measure theory is there term ring/algebra, but their definition are different and we can not deduce one from the other, the only requirement in definition they share is ...
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### Partition of R into midpoint convex sets

We say that a subset $X$ of $\mathbb{R}$ is midpoint convex if for any two points $a,b\in X$ the midpoint $\frac{a+b}{2}$ also lies in $X$. My question is: is it possible to partition $\mathbb{R}$ ...
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### Regular borel measures on metric spaces

When teaching Measure Theory last year, I convinced myself that a finite measure defined on the Borel subsets of a (compact; separable complete?) metric space was automatically regular. I used the ...
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### approximate a probability distribution by moment matching

Suppose we want to approximate a real-valued random variable $X$ by a discrete random variable $Z$ with finitely many atoms. Suppose all moments of $X$ is finite. We want to match the moments of $X$ ...
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### If a Borelian set has positive measure, does it contain a non empty open set (minus a measure null set)?

Let A be a borelian set with postivie measure. I was asking myself if it is possible to find an open set $B\subseteq A$ such that $B$ is an open set minus a set of null measure...
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### Is the sum of 2 Lebesgue measurable sets measurable?

Is the sum of two measurable set measurable? I think it is not...
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### Example for an integral, rectifiable varifold with unbounded first variation

I'm just looking for an example of an integral, rectifiable varifold, which has no locally bounded first variation. Recapitulation for every $m$-rectifiable varifold $\mu$ exists a $m$-rectifiable ...
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### Non Lebesgue measurable subsets with “large” outer measure

It is well known that for any set A in R^d there exists a measurable set E such that E contains A and m*(A)=m*(E). Is it possible to go the other direction? In other words, is it true that for any ...
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### Proving “almost all matrices over C are diagonalizable”.

This is an elementary question, but a little subtle so I hope it is suitable for MO. Let $T$ be an $n \times n$ square matrix over $\mathbb{C}$. The characteristic polynomial $T - \lambda I$ splits ...
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### Why are abelian groups amenable?

A (discrete) group is amenable if it admits a finitely additive probability measure (on all its subsets), invariant under left translation. It is a basic fact that every abelian group is amenable. ...
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### Measure between the counting measure and the Lebegue measure

There are subsets of the real line that has infinite counting measure, but Lebegue measure 0, so the Lebegue measure is used for measuring larger sets than the counting measure. My question is: Is ...
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### Suggestions for a good Measure Theory book

I have taken analysis and have looked at different measures, but I am currently looking at realizing a certain problem in a different light and feel that I need a better background in various measures ...
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### What's the use of a complete measure?

A complete measure space is one in which any subset of a measure-zero set is measurable. For what reasons would I want a complete measure space? The only reason I can think of is in the context of ...
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### Why is this generality in Vitali's Lemma useful?

In Vitali's Lemma it uses outer measure rather than measure. What are some results that depend on it this theorem applying to sets with only outer measure rather than measurable sets? Vitali's Lemma: ...
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### What are interesting families of subsets of a given set?

Motivation The usual starting point of both Topology and Measure Theory is the definition of a family of subsets of a set $S$. Indeed, one defines a topology on $S$ to be a family of subsets ...
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### What is the “continuity” in “absolute continuity”, in general?

The wikipedia article on absolute continuity gives a delta-epsilon definition for a measure $\mu$ defined on the Borel $\sigma$-algebra on the real line, with respect to the Lebesgue measure $\lambda$:...
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### Difference between Beta Process and Dirichlet process

I'm trying to understand the definition of a Beta process, as given in the paper: www.ece.duke.edu/~lcarin/Paisley_BP-FA_ICML.pdf The problem is that from the definition it follows that every ...
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### Definable collections of non measurable sets of reals

Is there a definable (in Zermelo Fraenkel set theory with choice) collection of non measurable sets of reals of size continuum? More verbosely: Is there a class A = {x: \phi(x)} such that ZFC proves "...