Let X be an arbitrary metric space and P be a porosity-like relation on X. We describe an infinite game which gives a characterization of σ-P-porous sets in X. This characterization can be applied to ordinary porosity above all but also to many other variants of porosity.

In his recent lecture at the International Congress [S], Stephen Semmes stated the following conjecture for which we provide a proof.Theorem. Suppose Ω is a bounded open set in Rn with n > 2, and suppose that B(0,1) ⊂ Ω, Hn-1(∂Ω) = M < ∞ (depending on n and M) and a Lipschitz graph Γ (with constant L) such that Hn-1(Γ ∩ ∂Ω) ≥ ε.Here Hk denotes k-dimensional Hausdorff measure and B(0,1) the unit ball in Rn. By iterating our proof we obtain a slightly stronger result which allows us...

This is a survey of various applications of the notion of the Choquet integral to questions in Potential Theory, i.e. the integral of a function with respect to a non-additive set function on subsets of Euclidean n-space, capacity. The Choquet integral is, in a sense, a nonlinear extension of the standard Lebesgue integral with respect to the linear set function, measure. Applications include an integration principle for potentials, inequalities for maximal functions, stability for solutions to...

We construct a Choquet simplex $K$ whose set of extreme points $T$ is $𝒦$-analytic, but is not a $𝒦$-Borel set. The set $T$ has the surprising property of being a $K_{\sigma \delta }$ set in its Stone-Cech compactification. It is hence an example of a $K_{\sigma \delta }$ set that is not absolute.

Let $G$ be a locally compact group. Let $L_t$ be the left translation in $L^{\infty }\left(G\right)$, given by $L_{t}f\left(x\right)=f\left(tx\right)$. We characterize (undre a mild set-theoretical hypothesis) the functions $f\in L^{\infty }\left(G\right)$ such that the map $t\to L_{t}f$ from $G$ into $L^{\infty }\left(G\right)$ is scalarly measurable (i.e. for $\phi \in L^{\infty }(G{)}^{*}$, $t\to \phi \left(L_{t}f\right)$ is measurable). We show that it is the case when $t\to \theta \left(L_{f}t\right)$ is measurable for each character $\theta$, and if $G$ is compact, if and only if $f$ is Riemann-measurable. We show that $t\to L_{t}f$ is Borel measurable if and only if $f$ is left uniformly continuous.Some of the measure-theoretic tools used there...

Let $X$ be a metric space with a doubling measure, $Y$ be a boundedly compact metric space and $u:X\to Y$ be a Lebesgue precise mapping whose upper gradient $g$ belongs to the Lorentz space $L_{m,1}$, $m\ge 1$. Let $E\subset X$ be a set of measure zero. Then ${\widehat{\mathscr{H}}}_m(E\cap u^{-1}\left(y\right))=0$ for ${\mathscr{H}}_m$-a.e. $y\in Y$, where ${\mathscr{H}}_m$ is the $m$-dimensional Hausdorff measure and ${\widehat{\mathscr{H}}}_m$ is the $m$-codimensional Hausdorff measure. This property is closely related to the coarea formula and implies a version of the Eilenberg inequality. The result relies on estimates of Hausdorff content of level sets...