# Tagged Questions

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

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### Reference request: Darboux properties of real-valued set functions (measures, densities, etc.)

Fix a set $S$ and let $f: \mathcal P(S) \rightharpoonup \mathbf R$ be a real-valued partial function on the power set of $S$; denote by $\mathcal D$ the domain of $f$. We say that $f$ has: (i) the ...
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### Riemannian Measures, Densities and Radon–Nikodym Theorem

If $M$ is a smooth manifold and $\mu$ is a $1$-density thereon then we may define a Borel measure (on Borel sets $A$) on $M$ as: $$\nu(A) = \int_M I_A \mu.$$ My question ...
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### interpret of Picone inequality for non-regular functions

Assume $\Omega \subset \mathbb{R}^N$, $N>4$ is open set. There is a well-known picone identity that says Let $u,v \in C^2(\Omega)$ satisfy $v>0$ and $-\Delta v \geq 0$ in $\Omega$. The ...
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### Relative null-ness

Here, "measure" always means Lebesgue measure on $\mathbb{R}$. This question is partly motivated by my answer http://math.stackexchange.com/questions/1444498/is-there-a-categorizaiton-system-for-null-...
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### Sets not containing the vertices of unit triangles (Question posed by Erdős)

Following this post, I have been thinking about the problem posed by Erdős, Does there exist a constant $c > 0$ such that every subset $A$ of the plane of area more than $c$ contains the ...
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### Multiplicity of a subcovering in spaces of given Hausdorff dimension

Let $X$ be a locally compact metric space of integer Hausdorff dimension $n$. Let $K\subset X$ be a compact subset. Let $\{B_i\}_i$ be a finite family of balls covering $K$. One may assume that all ...
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### Example of an adapted measurable process which is not Progressively Measurable

In this question Progressively measurable vs adapted, one finds a discussion on the subject of adapted processes versus progressively measurable processes. Counter-examples can be readily given. We ...
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### Can the integral of a “generic” bounded measurable function be determined by its values on the rationals?

[This question is an extension of my question Does a positive-measure subset of the unit interval almost surely intersect a random translation of some countable subgroup of $\mathbb{R}$?. I'm asking ...
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### “Lebesgue-measurable” cardinals and real-closed fields

I understand the motivation behind measurable cardinals is to ask the question: "is there any set large enough to admit a non-trivial measure on all of its subsets?" Hence, it's also worthwhile to ...
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### Measurability and Axiom of choice

In some situations, you need to show Lebesgue-measurability of some function on $\mathbb{R}^n$ and the verification is kind of lengthy and annoying, and even more so because measurability is "obvious" ...
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Let $\Delta^n$ be an $n$-simplex in $\mathbb{R}^n$. Let $V$ be the volume and $r$ the inradius (radius of the inscribed sphere) of $\Delta^n$. There is a well-known result that $$V \geq \frac{n^{n/2}(... 1answer 147 views ### Is it true that for dimension d=3, if v\in H_0^1(\Omega) but v\notin L_\infty(\Omega) then exponent of v or -v is not summable? I have the following Question: 1) Is it true that if \Omega\subset\mathbb R^3, \Omega - bounded, v\in H_0^1(\Omega) but v\notin L_\infty(\Omega) implies \int\limits_{\Omega}{e^vdx}=+\infty ... 0answers 139 views ### Equidistribution of spheres in \mathbb{R^2}/\mathbb{Z^2} Let \mathbb{H^2} be the hyperbolic upper half place, and let \Gamma be a lattice in SL(2,\mathbb{R}) acting on \mathbb{H^2}. A proof of the equidistribution of spheres on \mathbb{H^2/\Gamma} ... 2answers 198 views ### Is taking the product of signed measures weakly continuous? For a Polish space X, let C_b(X) denote the real Banach space of bounded continuous real-valued functions on X. Let M(X) denote the space of all finite signed Borel measures on X, equipped ... 0answers 103 views ### What does the Plancherel theorem say about positive-definite distributions? I'm trying to understand the answer to this MO question: Bochner's theorem for measures of positive type, which suggests a relationship between Bochner's theorem and the Plancherel theorem. The ... 2answers 291 views ### Commutative von Neumann algebras and localizable measure spaces This is not my subject so I apologize if my question is too obvious or understood from other pages. I read some pages such as Reference for the Gelfand-Neumark theorem for commutative von Neumann ... 2answers 396 views ### Is a specific sequentially closed subset of M([0,1]) closed? Let M([0,1]) be the set of finite signed measures on [0,1] (with the topology generated by the sets \left\{ \mu \in M([0,1]) : \left| \int f(x) \mu(dx)- a\right| \leq \delta\right\} for all \... 1answer 129 views ### Does a positive-measure subset of the unit interval almost surely intersect a random translation of some countable subgroup of \mathbb{R}? Please forgive me if this is a very easy question. Let A \subset [0,1] be a Borel-measurable set with strictly positive Lebesgue measure. Does there necessarily exist a countable subgroup G of \... 1answer 224 views ### continuity of the Boltzmann entropy in the Wasserstein metric For Lebesgue-absolutely continuous probability measures \rho\ll \mathcal{L}^d in the whole space \mathbb{R}^d with finite second moments (i-e \rho\in \mathcal{P}^2_{ac}(\mathbb{R}^d)), let$$ \...
Let $f : (\Omega, \mathcal F) \to (\mathbb R, \mathcal B(\mathbb R)$ be a measurable map, then it is well-known that $f$ could be approximated by a sequence $(f_n)$ of simple measurable functions, ...
### Sub-$\sigma$-algebras and conditional expectation
Is it true that any sub-$\sigma$-algebra of a Rokhlin-Lebesgue space is induced (up to completion) by a measurable map into another Rokhlin-Lebesgue space? In other words, is it true that conditional ...