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I am wondering if the Kolmogorov continuity theorem on Wikipedia is wrong? :

They say that the modification is sample continuous, and when we click on that link it says that it is a.s. continuous.

However, I've seen the same theorem beeing references in various books, and then they talk about a "continuous modification". But it seems then that the new process you get is not a-s continuous, but surely continuous. Can we assume that the modification is continuous everywhere, or do we have to be careful with it being discontinuous on a set of measure 0?

Michael Hardy
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user119615
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  • For anyone else wondering about this, I think I found an answer: If it is only a.s. continuous we just redefine it with an arbitrary continuous function where it is not continuous, this will still be a modification, because we only changed it on a set of measure 0. An hence we now have continuity surely. – user119615 Feb 12 '16 at 02:00

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