On the Skorokhod topology.(English)Zbl 0609.60005

The paper deals with the generalization of the Skorokhod space $$D[0,1]$$ for the case of functions taking values in a Hausdorff topological space $$E$$. The space $$D([0,1],E)$$ is the space of functions $$x\colon [0,1]\to E$$ which are right-continuous and admit left-hand limits for every $$t>0$$.
One of equivalent definitions of the Skorokhod topology in $$D([0,1],E)$$ is as follows. Let $$f\colon E\to \mathbb R$$ be a function. It induces a mapping $$\tilde f\colon D([0,1],E)\to D([0,1]\mathbb R)$$ by the formula $$(\tilde f(x))(t)= f(x(t))$$, $$t\in [0,1]$$. The Skorokhod topology on $$D([0,1],E)$$ is the coarsest topology with respect to which all $$\tilde f$$ are continuous, where $$f$$ runs through the set of all continuous functions from $$E$$ into $$\mathbb R$$. Narrower subclasses of continuous functions generating the same topology are considered as well.
The author investigates conditions on $$E$$ under which Borel and cylindrical $$\sigma$$-algebras coincide for $$D([0,1],E)$$. This is the case when $$E$$ is a separable metric space. Further the author finds tightness conditions for probability measures in the case when $$E$$ is a completely regular topological space. Analogous questions are considered for the space $$D({\mathbb{R}}^+,E)$$.

MSC:

 60B11 Probability theory on linear topological spaces
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