The assertion in question comes from the short final section in Theory of capacities of Choquet (1953/54), in connection with his prototype of the subsequent Choquet integral. The problem was whether and when this operation is additive. Choquet had the much more abstract idea that all functionals in a certain wide class must be subadditive, and similarly for superadditivity. His treatment of this point was more like an outline, and his proof limited to a rather narrow special case. Thus the proper...

A definition of “Šipoš integral” is given, similarly to [3],[5],[10], for real-valued functions and with respect to Dedekind complete Riesz-space-valued “capacities”. A comparison of Choquet and Šipoš-type integrals is given, and some fundamental properties and some convergence theorems for the Šipoš integral are proved.

We give a proof, based on the Poincaré inequality, of the symmetric property ($\tau$) for the Gaussian measure. If $f:{ℝ}^d\to ℝ$ is continuous, bounded from below and even, we define $Hf\left(x\right)={inf}_{y}f(x+y)+\frac{1}{2}{\left|y\right|}^2$ and we have$$\int {\mathrm{e}}^{-f}d{\gamma }_d\int {\mathrm{e}}^{Hf}d{\gamma }_d\le 1.$$This property is equivalent to a certain functional form of the Blaschke-Santaló inequality, as explained in a paper by Artstein, Klartag and Milman.

Let ${\varphi }_1,\cdots ,{\varphi }_n$ be real homogeneous functions in $C^{\infty }({ℝ}^n-\left\{0\right\})$ of degree $k\ge 2$, let $\varphi \left(x\right)=({\varphi }_1\left(x\right),\cdots ,{\varphi }_n\left(x\right))$ and let $\mu$ be the Borel measure on ${ℝ}^{2n}$ given by $$\mu \left(E\right)={\int }_{{ℝ}^n}{\chi }_E(x,\varphi \left(x\right)){\left|x\right|}^{\gamma -n}\mathrm{d}x$$ where $\mathrm{d}x$ denotes the Lebesgue measure on ${ℝ}^n$ and $\gamma >0$. Let $T_{\mu }$ be the convolution operator $T_{\mu }f\left(x\right)=(\mu *f)\left(x\right)$ and let $$E_{\mu }=\{(1/p,1/q)\parallel T_{\mu }{\parallel }_{p,q}<\infty ,1\le p,q\le \infty \}.$$ Assume that, for $x\ne 0$, the following two conditions hold: $det\left({\mathrm{d}}^2\varphi \left(x\right)h\right)$ vanishes only at $h=0$ and $det\left(\mathrm{d}\varphi \right(x\left)\right)\ne 0$. In this paper we show that if $\gamma >n(k+1)/3$ then $E_{\mu }$ is the empty set and if $\gamma \le n(k+1)/3$ then $E_{\mu }$ is the closed segment with endpoints $D=\left(1-\frac{\gamma }{n(k+1)},1-\frac{2\gamma }{n(k+1)}\right)$ and $D^{\text{'}}=\left(\frac{2\gamma }{n(1+k)},\frac{\gamma }{n(1+k)}\right)$. Also, we give some examples.

Let G be a group of homeomorphisms of a nondiscrete, locally compact, σ-compact topological space X and suppose that a Haar measure on X exists: a regular Borel measure μ, positive on nonempty open sets, finite on compact sets and invariant under the homeomorphisms from G. Under some mild assumptions on G and X we prove that the measure completion of μ is the unique, up to a constant factor, nonzero, σ-finite, G-invariant measure defined on its domain iff μ is ergodic and the G-orbits of all points...

Dans ces notes il sera expliqué que la propriété $MCP(0,5)$ est vérifiée par le groupe de Heisenberg ${ℍ}^1$ muni de la distance de Carnot-Carathéodory et de la mesure de Lebesgue. Cette propriété correspond pour les espaces métriques mesurés à une courbure de Ricci positive. Comme application, les mesures interpolées par transport de mesure sont absolument continues. En revanche, la courbure-dimension $CD(0,N)$, une autre courbure de Ricci synthétique adaptée aux espaces métriques mesurés est fausse pour ${ℍ}^1$.