Talk:Law (stochastic processes)

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Latest comment: 5 years ago3 comments2 people in discussion

Nothing is said for now about the choice of a sigma-field on the set of functions. This is a somewhat subtle matter! The minimal sigma-field, generated by evaluations, fails to make the set of continuous functions measurable in the Wiener measure case. Boris Tsirelson (talk) 12:15, 12 September 2010 (UTC)Reply

This has been bothering me as well; I notice you have been tackling this more recently in the main Stochastic Process article. I'm not sure, but there may be hints at a resolution in Appendix A (Section A.1.1) of http://page.math.tu-berlin.de/~friz/master4_May6th.pdf. Usernames are difficult (talk) 18:04, 30 January 2019 (UTC)Reply

Yes, this is more or less standard "resolution", and as such, worth mentioning on Wikipedia.

However, 50 years ago I (being young) felt unsatisfied with it. You see, the set of continuous functions is (for Wiener measure) of full outer measure. But the set of everywhere discontinuous function is, too. Isn't it subjective, to prefer the former to the latter?

You may say: do not be ridiculous; we all understand that continuity is good, and discontinuity is bad. True. But consider the Poisson measure (for the Poisson process that jumps by 1 at random instants). Now continuous functions are of zero outer measure. Right continuous functions are of full outer measure. Left continuous functions are of full outer measure, too. What now? Which option is ridiculous?..

This dissatisfaction led me to a research mentioned briefly in Note 6 to "Baire set" article. See also Ron Peled, if you feel interested. Boris Tsirelson (talk) 20:44, 30 January 2019 (UTC)Reply

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