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Probability measure on smooth functions 
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#1
Oct610, 12:00 AM

P: 810

Is there a "standard" probability measure one would use for the set of smooth realvalued functions on [a, b]?
My intuition is picturing a setup where you cut out shapes in the xy plane, and then the set of functions whose graphs are contained in that shape have a measure proportional to the Euclidean area of the shape. But I can't quite make that intuition exact. 


#2
Oct710, 06:10 AM

P: 336

Do you have the Borel measure ( under the sup metric ) in mind?



#3
Oct810, 10:16 AM

P: 9

I suppose, you have to consider functions uniformely bounded by some constant M (or even vith uniformely bounded variation?), otherwise the whole set gets infinite measure, not 1, the way you described the measure.



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