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.
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.