Generalized Limit Expressed through Ultralimits

In my book Limit of a Discontinuous Function I defined the generalized limit defined for every (even discontinuous) function.

The definition of my generalized limit uses the concept of funcoids. Funcoids are a little advanced topic, what somehow hinder understanding of generalized limit by other mathematicians.

But in the special case of a function p -> q where p is a vector topological space (for example, a normed space, Banach space, Hilbert space, etc.) and q is a compact topological space, we can equivalently define the generalized limit without using the concept of funcoids.

First, let me remind the definition of ultralimit as defined in the “standard” nonstandard analysis:

For convenience, the set of filters will include the improper filter.

We can define the image by a function f of a filter X as the minimal filter that contains all images of elements of X.

Ultralimit of a function f regarding an ultrafilter (I allow any ultrafilter here, even a principal one, for convenience) a is the limit point of the image of a by f.

If the space q is compact, then then the ultralimit always exists because the image of a by f is also an ultrafilter and the well-known fact in a compact space every ultrafilter has a limit point.

The definition of generalized limit to a compact space without using funcoids follows:

Let a be some point of the space q. Consider the function t that maps every ultrafilter convergent to a into its ultralimit. (Exercise: define the corresponding concept for a limit of a sequence.)

It remains the last step: consider the set of compositions of t with all possible shifts in our vector space a. This is what I call the generalized limit.

Why these looking random operations? Read my book.

This is equivalent to the definition of the generalized limit in my book, because funcoid is determined by its values on ultrafilters.

The generalized limit has the following properties:

  • It exists for every function (in our special case with the requirement of compactness of the destination space, but the equivalent definition in my book does not require compactness).
  • The limit at a point in the usual sense is determined by the generalized limit (because we include the principal ultrafilter determined by a).
  • Having a function on the space b we can extend this function to a function acting on values of generalized limit (exercise).
  • The operations on the values of generalized limits follow the usual algebra laws. For example if the space b is real numbers or complex numbers or a vector space, then yy = 0 for every value y of the generalized limit.

So far, we defined the generalized limit to compact spaces. See my book Limit of a Discontinuous Function for a generalized case if b is not necessarily compact.

Having the generalized limit, we can easily define the derivative of an arbitrary function or the definite integral of an arbitrary function!

See my book on how to define nondifferentiable solutions of differential equations. It is a new science.

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