We formulate a Covering Property Axiom $CP{A}_{cube}$, which holds in the iterated perfect set model, and show that it implies easily the following facts. (a) For every S ⊂ ℝ of cardinality continuum there exists a uniformly continuous function g: ℝ → ℝ with g[S] = [0,1]. (b) If S ⊂ ℝ is either perfectly meager or universally null then S has cardinality less than . (c) cof() = ω₁ < , i.e., the cofinality of the measure ideal is ω₁. (d) For every uniformly bounded sequence $⟨fₙ\in {ℝ}^{ℝ}{⟩}_{n<\omega }$ of Borel functions there are sequences:...

Pour trois fonctions non-négatives intégrables sur ${\mathbf{R}}^n$, $f,g$ et $h$, telles que $\left(h\right(x+y){)}^{-1/n}\le (f\left(x\right){)}^{-1/n}+\left(g\right(y){)}^{-1/n}$, Borelll a établi l’inégalité $\int h\left(z\right)dz\ge \min \int f\left(x\right)dx,\int g\left(y\right)dy)$. Nous déterminons les conditions précises où l’inégalité sera stricte. La clef de cette analyse est une nouvelle caractérisation des fonctions convexes mesurables.

We give a complete characterization of those $f:[0,1]\to X$ (where $X$ is a Banach space) which allow an equivalent $C^{1,\mathrm{BV}}$ parametrization (i.e., a $C^1$ parametrization whose derivative has bounded variation) or a parametrization with bounded convexity. Our results are new also for $X={ℝ}^n$. We present examples which show applicability of our characterizations. For example, we show that the $C^{1,\mathrm{BV}}$ and $C^2$ parametrization problems are equivalent for $X=ℝ$ but are not equivalent for $X={ℝ}^2$.