# Probability measure on smooth functions

• Tac-Tics
In summary, the conversation discusses the possibility of using a "standard" probability measure for the set of smooth real-valued functions on [a, b]. One suggests using a measure proportional to the Euclidean area of the shape formed by cutting out shapes in the x-y plane, while the other brings up the Borel measure under the sup metric and the need to consider functions uniformly bounded by a constant to avoid infinite measure.
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.

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

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.

## 1. What is a probability measure on smooth functions?

A probability measure on smooth functions is a mathematical concept used to describe the likelihood of certain events occurring in a smooth function. It assigns a numerical value between 0 and 1 to each event, with 0 representing impossibility and 1 representing certainty.

## 2. How is a probability measure on smooth functions different from a traditional probability measure?

A probability measure on smooth functions differs from a traditional probability measure in that it is specifically applied to smooth functions, which are functions that have derivatives of all orders. This allows for a more precise and detailed analysis of probabilities within a continuous domain.

## 3. What are some common applications of probability measure on smooth functions?

Probability measure on smooth functions has various applications in fields such as physics, engineering, finance, and statistics. It is often used to model and predict the behavior of complex systems and to make informed decisions based on uncertain outcomes.

## 4. How is a probability measure on smooth functions calculated?

A probability measure on smooth functions is calculated using integration techniques, where the probability of an event is represented by the area under a curve on a smooth function. This can be done using various mathematical tools such as the Lebesgue integral or the Riemann integral.

## 5. Can probability measure on smooth functions be used for non-smooth functions?

No, probability measure on smooth functions is specifically designed for smooth functions and cannot be directly applied to non-smooth functions. However, there are techniques such as regularization that can be used to approximate non-smooth functions and apply probability measure on them.

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