Probability measure on smooth functions

  1. 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.
  2. jcsd
  3. Do you have the Borel measure ( under the sup metric ) in mind?
  4. 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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