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Probability measure on smooth functions

by Tac-Tics
Tags: functions, measure, probability, smooth
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Tac-Tics
#1
Oct6-10, 12:00 AM
P: 810
Is there a "standard" probability measure one would use for the set of smooth real-valued functions on [a, b]?

My intuition is picturing a setup where you cut out shapes in the x-y 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.
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Eynstone
#2
Oct7-10, 06:10 AM
P: 336
Do you have the Borel measure ( under the sup metric ) in mind?
simeonsen_bg
#3
Oct8-10, 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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