We start with a metrizable topological space. There are many equivalent
axiomatizations of a metrizable topology; _e.g._, see Kelley. Perhaps
the easiest way to visualize a metrizable topology is to consider that
one was given a metric space but that he lost his notes in which the
exact form of the metric was written down. Thus one knows that he can
do everything that he could in a metric space, if only he can figure
out how.
The “figuring out how” is by no means trivial. Here, it will be assumed
that a cumulative probability distribution has been obtained on the
space by one of the standard methods; bird in cage,[11] Munroe I,[12]
Munroe II,[13] ordering (see Halmos[14] or Kelley[15]). This cumulative
probability distribution is a function on X onto the interval [0,1] of
real numbers. The inverse of this function, which exists by the Radon
Nikodym theorem, provides a mapping from the real interval onto the
non-trivial portion of X. This mapping induces all of the pleasant
properties of the real numbers on the space X: topological, metric, and
ordering.
Actually, it turns out that, especially if the dimensionality of
the space is greater than one, the foregoing procedure not only
provides one metrization, but many. Indeed, this lack of uniqueness
is what makes the procedure exceedingly difficult. Only by imposing
some additional conditions that result in the existence of a unique
solution, does the problem become tractable.
We choose to impose the additional condition that the resulting metric
space be a Euclidean geometry with a rectangular coordinate system.
[11] Harman, W. W., “Principles of the Statistical Theory of
Communication,” New York, New York:McGraw-Hill, 1963.
[12] Munroe, M. E., “Introduction to Measure and Integration,”
Cambridge, Mass.:Addison-Wesley, 1953.
[13] Munroe, M. E., “Introduction to Measure and Integration,”
Cambridge, Mass.:Addison-Wesley, 1953.
[14] Halmos, P. R., “Measure Theory,” Princeton, New Jersey:D. Van
Nostrand Co., Inc., 1950.
[15] Kelley, J. L., “General Topology,” Princeton, New Jersey:D. Van
Nostrand Co., Inc., 1955.
Even this always does not yield uniqueness, but we will show the
additional restriction that will guarantee uniqueness after the
necessary language is developed. Since all metrizations of a given
metrizable topology are isomorphic, in the quotient class the
orthogonal Euclidean geometry serves the purpose of being a convenient
representative of the unique element resulting from a given metrizable
topology.
Furthermore, the same comment applies to the use of a Gaussian
distribution as the probability distribution on this orthogonal
Euclidean geometry. Namely, the random Gaussian distribution on an
orthogonal Euclidean geometry is a convenient representative member of
the equivalence class which maps into one element (stochastic space) of
the quotient class.