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**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 \(D({\mathbb{R}}^+,E)\).

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 \(D({\mathbb{R}}^+,E)\).

Reviewer: S. A. Chobanyan (Tbilisi)

### MSC:

60B11 | Probability theory on linear topological spaces |