For the functor of upper semicontinuous capacities in the category of compact Hausdorff spaces and two of its subfunctors, we prove open mapping theorems. These are counterparts of the open mapping theorem for the probability measure functor proved by Ditor and Eifler.

In this paper, motivated by questions in Harmonic Analysis, we study the operation of (countable) increasing union, and show it is not idempotent: ${\omega }_1$ iterations are needed in general to obtain the closure of a class under this operation. Increasing union is a particular Hausdorff operation, and we present the combinatorial tools which allow to study the power of various Hausdorff operations, and of their iterates. Besides countable increasing union, we study in detail a related Hausdorff operation,...

Let $ℬ=f:ℝ\to ℝ:fisbounded$, and $ℳ=f:ℝ\to ℝ:fisLebesguemeasurable$. We show that there is a linear operator $\Phi :ℬ\to ℳ$ such that Φ(f)=f a.e. for every $f\in ℬ\cap ℳ$, and Φ commutes with all translations. On the other hand, if $\Phi :ℬ\to ℳ$ is a linear operator such that Φ(f)=f for every $f\in ℬ\cap ℳ$, then the group $G_{\Phi }$ = a ∈ ℝ:Φ commutes with the translation by a is of measure zero and, assuming Martin’s axiom, is of cardinality less than continuum. Let Φ be a linear operator from ${ℂ}^{ℝ}$ into the space of complex-valued measurable functions. We show that if Φ(f) is non-zero for every $f\left(x\right)=e^{cx}$, then $G_{\Phi }$ must...

In this paper, we study optimal transportation problems for multifractal random measures. Since these measures are much less regular than optimal transportation theory requires, we introduce a new notion of transportation which is intuitively some kind of multistep transportation. Applications are given for construction of multifractal random changes of times and to the existence of random metrics, the volume forms of which coincide with the multifractal random measures.

Let T be a bounded linear operator on a (real or complex) Banach space X. If (aₙ) is a sequence of non-negative numbers tending to 0, then the set of x ∈ X such that ||Tⁿx|| ≥ aₙ||Tⁿ|| for infinitely many n’s has a complement which is both σ-porous and Haar-null. We also compute (for some classical Banach space) optimal exponents q > 0 such that for every non-nilpotent operator T, there exists x ∈ X such that $\left(\right|\left|Tⁿx\right||/\left|\right|Tⁿ\left|\right|)\notin {\ell }^q\left(\mathbb{N}\right)$, using techniques which involve the modulus of asymptotic uniform smoothness of X.

Let $𝔐$ and $\Re$ be algebras of subsets of a set $\Omega$ with $𝔐\subset \Re$, and denote by $E\left(\mu \right)$ the set of all quasi-measure extensions of a given quasi-measure $\mu$ on $𝔐$ to $\Re$. We give some criteria for order boundedness of $E\left(\mu \right)$ in $ba\left(\Re \right)$, in the general case as well as for atomic $\mu$. Order boundedness implies weak compactness of $E\left(\mu \right)$. We show that the converse implication holds under some assumptions on $𝔐$, $\Re$ and $\mu$ or $\mu$ alone, but not in general.

Let $X$ be a completely regular Hausdorff space, $E$ a boundedly complete vector lattice, $C_b\left(X\right)$ the space of all, bounded, real-valued continuous functions on $X$, $ℱ$ the algebra generated by the zero-sets of $X$, and $\mu C_b\left(X\right)\to E$ a positive linear map. First we give a new proof that $\mu$ extends to a unique, finitely additive measure $\mu ℱ\to E^{+}$ such that $\nu$ is inner regular by zero-sets and outer regular by cozero sets. Then some order-convergence theorems about nets of $E^{+}$-valued finitely additive measures on $ℱ$ are proved, which extend...

Let $𝔐$ and $\Re$ be algebras of subsets of a set $\Omega$ with $𝔐\subset \Re$, and denote by $E\left(\mu \right)$ the set of all quasi-measure extensions of a given quasi-measure $\mu$ on $𝔐$ to $\Re$. We show that $E\left(\mu \right)$ is order bounded if and only if it is contained in a principal ideal in $ba\left(\Re \right)$ if and only if it is weakly compact and $extrE\left(\mu \right)$ is contained in a principal ideal in $ba\left(\Re \right)$. We also establish some criteria for the coincidence of the ideals, in $ba\left(\Re \right)$, generated by $E\left(\mu \right)$ and $extrE\left(\mu \right)$.