If I remember correctly, what you are describing is close to the concept of a distribution function (it is the name that I am slightly unsure about).
Given a measure space (X,E,μ), and a measurable function f:X→R, a concept often used in classical and harmonic analysis is the notion of the distribution function λf:R+→R defined as
λf(a)=μ({|f|<a})
that is, λf is returns the measure of the set on which |f| is bounded by a. One can reformulate the Lp norms of f in terms of the distribution function
∥f∥Lp(X,μ)=p∫∞0tp−1λf(t)dt
a device that is very useful for analytical estimates. Now, formally speaking the functional L(f) you wrote down is very similar to the derivative λ′f.
Another possibility that you may want to look into is the notion of pull-back distributions. If f:M→R is a smooth map with non-vanishing gradient, then given any distribution (generalized function) k on R
you can define the pull-back distribution f∗k on M. The definition of f∗k depends closely on the notion of L(f) you wrote down (see e.g. the introduction to distributions by Friedlander and Joshi), and is a generalization of the co-area formula.
I am not exactly sure how to answer your second question of "is there a good way to compute". If you are asking whether given an analytic expression for the function f whether you can have a "closed form" expression for L(f), the answer is surely no, since the expression also depends on the chosen Riemannian metric/volume form/measure on M. If you are asking whether there are any convenient expressions to capture L(f), I think you need to say a bit more about why you care about L(f). You can certainly just write L(f)=f∗δa as the pull-back of the Dirac delta distribution on R; whether that expression is a useful one for your purpose, I don't know.
This post imported from StackExchange Mathematics at 2014-06-02 20:26 (UCT), posted by SE-user Willie Wong