Zero-Measure Equivalence Classes

The whole structure we are examining here induces one to think that it would be a reasonable thing to do if we decided to group all the real functions into zero-measure equivalence classes. We could consider as equivalent, at least from the point of view of the physics applications, two real functions which differ by a zero-measure function. For this purpose, a zero-measure real function is a function whose absolute integral over $[-\pi,\pi]$ is zero, or that has zero integral on any closed sub-interval of the interval $[-\pi,\pi]$.

Then one would consider in a group all functions that are zero measure in this sense, such as the identically null function. This class could then be represented simply by that particular element, $f(\theta)\equiv0$, which is quite clearly the smoothest element within that class. Given any real function with non-zero measure, all other functions that differ from it by a zero-measure function would be in the same equivalence class. Instead of considering all real functions, we could formulate everything that was discussed here in terms of these equivalence classes.

Since the Fourier coefficients are defined by integrals, it is immediately clear that a given sequence of Fourier coefficients, if it belongs to any real function at all, belongs to one such equivalence class, rather than to the individual functions. Therefore, there is also some mathematical sense to such a classification. The proposed method of representation of the real functions, as limits to the unit circle of real or imaginary parts of inner analytic functions within the open unit disk, gives us then a clear definition of a preferred or central element of each class: that particular real function which is obtained at the limit just described. Being a limit from a perfectly smooth analytic function, this is clearly the smoothest element of the class.

One could consider this classification process, and the representation of each zero-measure equivalence class by its smoothest element, as a process of elimination of what we might call the irrelevant pathologies of the real functions. The only pathologies that would remain would be those that have a definite effect on the integral of the function. This certainly makes sense in terms of the physics applications, but it might be of some intrinsic mathematical interest as well.