Probability measure on smooth functions

[Tac-Tics]

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.

 

[Eynstone]

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

 

[simeonsen_bg]

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.