Náhodné měřitelné množiny a procesy částic

Jurčo, Adam

Random measurable sets and particle processes Adam Jurčo Abstract In this thesis we deal with particle processes on more general spaces. First we in- troduce the space of Lebesgue measurable sets represented by indicator functions with topology given by L1 loc convergence. We the explore the topological properties of this space and its subspaces of sets of finite and locally finite perimeter. As these spaces do not satisfy the usual topological assumptions needed for construction of point processes we use another approach based on measure-theoretic assumptions. This will allow us to define point processes given by finite dimensional distributions on measurable subsets of the space of Lebesgue-measurable sets. Then we will derive a formula for a volume fraction of a Boolean process defined in this more general setting. Further we introduce a Boolean process with particles of finite perimeter and derive a formula for its specific perimeter. 1

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