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Continuity properties of sample functions

  • Kai Lai Chung
Part of the Die Grundlehren der Mathematischen Wissenschaften book series (volume 104)

Abstract

In what follows we shall give further theorems concerning almost all (3) sample functions. It is important to distinguish between two kinds of assertions: (i) an assertion about x (t, ω) for a. a.ω at a fixed t which is the same for allω; (ii) an assertion about x(•,ω ) for a. a.ω at a generic regarded as the running adscissa in the sample graph (t,x(t, ω) ),t∉T,for each ω. thus the exceptional null set may depend on t in case (i) but not in case(ii) in Naturally (i) is a special case of (ii). As anc an example, the second assertion in Theorem 5.6 may be stated as fol- lows: for a fixed t and stable i, t∉ S i (w) implies that t is in an (open) i-interval of x(•, ω) for a.a. w. This is an assertion of the first kind; the corresponding assertion of the second kind is false at a generic t which is an endpoint of an i-interval. Similarly, Theorem 2 below is for an assertion of the first kind while the corresponding assertion of the second kind will be given in Theorem 7.4.

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Copyright information

© Springer-Verlag OHG. Berlin · Göttingen · Heidelberg 1960

Authors and Affiliations

  • Kai Lai Chung
    • 1
  1. 1.Syracuse UniversityUSA

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