This note contains a proof of the existence of a one-to-one function of onto itself with the following properties: is a rational-linear automorphism of , and the graph of is a non-measurable subset of the plane.
We state and prove a stability result for the anisotropic version of the isoperimetric inequality. Namely if is a set with small anisotropic isoperimetric deficit, then is “close” to the Wulff shape set.
S. Solecki proved that if is a system of closed subsets of a complete separable metric space , then each Suslin set which cannot be covered by countably many members of contains a set which cannot be covered by countably many members of . We show that the assumption of separability of cannot be removed from this theorem. On the other hand it can be removed under an extra assumption that the -ideal generated by is locally determined. Using Solecki’s arguments, our result can be used...
This essay outlines a generalized Riemann approach to the analysis of random variation and illustrates it by a construction of Brownian motion in a new and simple manner.
Let X be a completely regular Hausdorff space, E and F be Banach spaces. Let Cb(X, E) be the space of all E-valued bounded, continuous functions on X, equipped with the strict topology β. We develop the Riemman-Stieltjes-type Integral representation theory of (β, || · ||F) -continuous operators T : Cb(X, E) → F with respect to the representing Borel operator measures. For X being a k-space, we characterize strongly bounded (β, || · ||F)-continuous operators T : Cb(X, E) → F. As an application, we...
We introduce and study a rough (approximate) curvature-dimension condition for metric measure spaces, applicable especially in the framework of discrete spaces and graphs. This condition extends the one introduced by Karl-Theodor Sturm, in his 2006 article On the geometry of metric measure spaces II, to a larger class of (possibly non-geodesic) metric measure spaces. The rough curvature-dimension condition is stable under an appropriate notion of convergence, and stable under discretizations as...
The Monge-Kantorovich problem is revisited by means of a variant of the saddle-point method without appealing to c-conjugates. A new abstract characterization of the optimal plans is obtained in the case where the cost function takes infinite values. It leads us to new explicit sufficient and necessary optimality conditions. As by-products, we obtain a new proof of the well-known Kantorovich dual equality and an improvement of the convergence of the minimizing sequences.
The Monge-Kantorovich problem is revisited by means of a variant of the saddle-point method without appealing to c-conjugates. A new abstract characterization of the optimal plans is obtained in the case where the cost function takes infinite values. It leads us to new explicit sufficient and necessary optimality conditions. As by-products, we obtain a new proof of the well-known Kantorovich dual equality and an improvement of the convergence of the minimizing sequences.
We find a condition for a Borel mapping which implies that the Hausdorff dimension of is less than or equal to m-n for almost all .