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

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### Support of a regular measure Reg

Let $K$ be compact, Hausdorff space but not necessarily metrizable. Let $\mathfrak{M}$ be the Borel $\sigma$ field over $K$ and $\mu$ be a positive Regular Borel measure on $K$. Let $S$ be a subset of ...

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46 views

### Integration on a family of differential forms

Let $X$ be a smooth manifold, and denote by $\Omega^*(X)$ the set of all smooth differential forms on $X$. Assume we have a family of differential forms $\omega_t \in \Omega^*(X)$, $t\in E$, ...

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106 views

### Minkowski sum of polytopes from their facet normals and volumes

By Minkowski's work in the early 1900s, every polytope $P\subset\mathbb R^n$ is determined up to translation by its unit facet normals $u_1,\dots,u_k$ and facet volumes $\alpha_1,\dots,\alpha_k$. ...

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### Egorov's and Lusin's Theorem in the space with infinite measure

Both the fundamental Egorov's and Lusin's Theorem in measure theory are given on any measurable space $X$ whose measure is finite.
On the measurable space whose measure is infinite, does there ...

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34 views

### Monotonicity given an implicit function containing a Measure integral

The following question seems simple but I am not sure how to handle it correctly because of the integral with respect to a measure. I would be very thankful for any reply.Cheers!
Knowing that $$f(\...

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62 views

### Regularity in Orlicz spaces for the Poisson equation

I see the following Lemma in :Regularity in Orlicz spaces for the Poisson equation | SpringerLink:(2007)
$$\Delta u=f \quad \quad \quad \quad (1)$$
Lemma 2: There is a constant $N_1 >1$ so that ...

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202 views

### Effect of perturbing the atoms of a measure on the Wasserstein distance

Let $(X,d)$ be a metric space, $x_1,\ldots,x_N\in X$ and $x_1',\ldots,x_N'\in X$ be atoms, and $G=\sum_{i=1}^Np_i\delta_{x_i}$, $G'=\sum_{i=1}^Np_i'\delta_{x_i}$, and $G''=\sum_{i=1}^Np_i'\delta_{x_i'}...

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128 views

### continuity entropy with respect gibbs measures

Let $X=\{0,1\}^{\mathbb{N}}$. For simplicity I consider Bernoulli measures on $X$ only.
Let $f:X\to \mathbb{R}$ be Holder continuous. The measure $\mu$ is a Gibbs measure with potential $f$ if there ...

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99 views

### Completely I-non-measurable unions in Polish spaces

Problem. Let $X$ be a Polish space, $\mathcal I$ be a $\sigma$-ideal with Borel base, and $\mathcal A\subset\mathcal I$ be a point-finite cover of $X$. Is it true that $\mathcal A$ conatins a ...

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### Measure preserving coordinates of $S^2$ from $[0,1]^2$

Consider the unit sphere $S^2 = \left\{x\in\mathbb{R}^3 ~ {\large|} ~ |x|=1 \right\}$ and denote the uniform (Lebesgue) measures on the $S^2$ and $[0,1]^2$ by $m_S$ and $m_2$, respectively.
Question ...

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### Weak* convergence in a dual Banach lattice vs norm convergence of moduli

Let $E$ be a dual Banach lattice, that is, $E = E_*^*$ for some Banach lattice $E_*$ (I have $M(X)=C(X)^*$ specifically in mind for a compact space $X$).
Suppose that $(x_n)$ is a weak*-null ...

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162 views

### Weak*-convergence of signed measures

Let $X$ be a compact Hausdorff space and let $M(X)$ denote the space of signed measures that is naturally dual to $C(X)$, the space of continuous functions on $X$. I am interested whether the ...

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86 views

### Borel $\sigma$-algebra in $\beta \mathbb N \times \beta \mathbb N$

For a second-countable space $X$, we have $${\rm Bor}(X\times X) = {\rm Bor}\, X \otimes {\rm Bor}\, X,$$ that is the Borel $\sigma$-algebra of the product is the product $\sigma$-algebra. Some ...

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190 views

### Is there a finitely additive measure on R which is not sigma-additive?

Consider the usual measurable space of real number $( \mathbb{R}, \mathcal{B}(\mathbb{R}))$.
My question is:
Is there an application $\mu$ on $\mathcal{B}( \mathbb{R}) \rightarrow [0,+\infty]$ such ...

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434 views

### Is taking the positive part of a measure a continuous operation?

Here is question I tried to answer for some time - it seems to be straightforward, but I have trouble figuring it out.
Let $\Omega$ be a compact domain in $\mathbb{R}^n$. For any signed Borel measure ...