Continuity properties of sample functions

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.