Part 14

Munroe, M. E., “Introduction to Measure and Integration,” Cambridge, Mass.:Addison-Wesley, 1953.

Halmos, P. R., “Measure Theory,” Princeton, New Jersey:D. Van Nostrand Co., Inc., 1950.

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.

Information Theory

Now, we will show that Information Theory provides the language necessary to describe the metrization procedure in detail.

It is possible to introduce Information Theory axiomatically by a suitable generalization of the axioms in Feinstein. But to simplify the discussion here, we will use the less elegant but equivalent method of defining certain definite integrals. The probability density distribution p is defined from the cumulative probability distribution P by

P(X′) = ∫X′_{measurable ⊂ X} p(x)dx. (1)

Then the information rate H is defined as

H(X) = -∫ₓp(x) ln κ p(x)dx (2)

where kappa has (carries) the units of X. Finally, the channel rate R is defined as

R(⨀Xᵢ) = ΣH(Xᵢ) - H(X), (3) I I

where X is the denumerable cartesian product space

X = ⨂Xᵢ. (4) I

Feinstein uses his axioms only in finite space X; i.e., card(X) < K₀.

Feinstein, A., “Foundations of Information Theory,” New York, New York: McGraw-Hill, 1958.

If I is infinite, certain precautions have to be exercised.

Next, we define the angle Θ

|Θ(⨀Xᵢ)| = sin⁻¹e^{-R(⨀Xᵢ)} (5) I and the norm

|X| = κ(2πe)⁻¹ᐟ² e^{(HX)}. (6)

Now, if a statistically independent basis; i.e., one for which κ R(⨀Xᵢ) ≡ constant, (7) I

can be provided in terms of one-dimensional components; i.e., none of them can be decomposed further, then it is just the usual problem of diagonalization of a symmetric matrix by means of a congruence transformation to provide an orthogonal coordinate system. Furthermore, for uniqueness, we arrange the spectrum in decreasing order. Then, by means of the Radon Nikodym theorem applied to each of these one-dimensional axes, the probability distribution may be made; e.g., Gaussian, if desired. Thus, we obtain the promised orthogonal Euclidean space.

This “if” is the catch that makes all methods of metrization of a space of dimensionality higher than one impractical, except the method of successive projections upon unit spheres centered at the center of gravity. The method of using that nilpotent projection operator is described in the companion paper(see footnote page 65).

Channel

At this time we can state the remaining additional condition required that a decomposition be unique. The index space I has to be partitioned into exactly two parts, say I′ and I″; i.e.,

I′ ∪ I″ = I (8)

I′ ∩ I″ = φ,