Topology and geometry of spaces of measures
A talk in the Keine Reihe series by
Vladimir Bogachev from Moscow State University
| Abstract: | We shall discuss some basic topological and metric properties of spaces of measures
connected with weak convergence. A standard fact is that the space of probability
measures on a complete separable metric space can be equipped with diverse complete
separable metrics
(such as Prohorov or Kantorovich) generating the weak topology. However, the whole space
of signed bounded measures with the weak topology is not metrizable in nontrivial
cases in spite of the fact that the Kantorovich norm is defined on the whole space.
This phenomenon will be discussed in detail
along with a number of elementary examples
(such as measures on $\mathbb{N}$, $[0,1]$ and $\mathbb{R}$), which turn out
rather subtle in the case of signed measures. The lecture does not assume any
special knowledge except for some basic concepts related to metric spaces and measures,
but even at this level there are interesting questions for future research. |