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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
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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
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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.
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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
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outer measure. Left continuous functions are of full outer measure, too. What now? Which option is ridiculous?..
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Yes, this is more or less standard "resolution", and as such, worth mentioning on
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This dissatisfaction led me to a research mentioned briefly in Note 6 to "
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http://page.math.tu-berlin.de/~friz/master4_May6th.pdf
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