Measuring the uniformity of Measurable Subsets of The Unit Square

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Measuring the uniformity of Measurable Subsets of The Unit Square

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Bharath Krishnan^*

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Submitted:

11 July 2023

Posted:

12 July 2023

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Abstract

Suppose set

Keywords:

Subject: Computer Science and Mathematics - Mathematics

0. Intro

The aim of this paper is to measure the "uniformity" of measurable subsets of

{[0, 1]}^{2}

. If set

A \subseteq [0, 1] \times [0, 1]

; we want to define a measure of uniformity for A.

Here are some example of a "uniform A":

(1): $A_{n}$ is the uniform distribution on the uniform grid $G_{n} : = \{(\frac{i}{n}, \frac{j}{n}) : i = 0, \dots, n, j = 0, \dots, n\}$ such that with large n, if $A_{n} \to A$ , then A is uniform in ${[0, 1]}^{2}$
(2): For all real $x_{1}, x_{2}, y_{1}, y_{2}$ , if $0 \leq x_{1} < x_{2} \leq 1$ and $0 \leq y_{1} < y_{2} \leq 1$ where the Lebesgue measure (on the Lebesgue sigma-algebra) of $([x_{1}, x_{2}] \times [y_{1}, y_{2}]) \cap A$ is $(x_{2} - x_{1}) (y_{2} - y_{1})$ , then set A is uniform in $[0, 1] \times [0, 1]$ .

(In general, we shall define a "uniform" subset of

{[0, 1]}^{2}

in def. 4,

§

2,).

Note we wish for a measure of uniformity to be between (and including) zero and one or zero and infinity such that the larger the measure, the larger the non-uniformity.

Further note, there are already several measures of uniformity for finite points in the unit square (e.g. wasserstein distance [1] or distance between empirical copula & independence copula [2]) but no measure for infinite points in the unit square.

1. Preliminary Definitions

Definition 1

(Hausdorff Measure). Let

(X, α)

be a metric space,

α \in [0, \infty)

. For every

C \in X

, define the diameter of C as:

diam (C) : = sup \{φ (x, y) : x, y \in C\}, diam (\emptyset) : = 0

i \in N

and

δ \in R

such that

δ > 0

, where the Euler’s Gamma function is Γ and constant

N_{α}

is:

N_{α} = \frac{π^{α / 2}}{Γ (\frac{α}{2} + 1)}

(1.0.1)

we define:

H_{δ}^{α} (E) = N_{α} inf \{\sum_{i = 1}^{\infty} {(diam (C_{i}))}^{α} : diam (C_{i}) \leq δ, E \subseteq ⋃_{i = 1}^{\infty} C_{i}\}

(1.0.2)

such if the infimum of the equation is taken over the countable covers of sets

C_{i} \subset X

of E (satisfying

diam (C_{i}) \leq δ

), the Hausdorff Outer Measure is:

H^{α} (E) = sup_{δ > 0} H_{δ}^{α} (E) = lim_{δ \to 0} H_{δ}^{α} (E)

where for

α \in N

H^{α} (E)

coincides with the α-dimensional Lebesgue Measure, where we convert the Outer measure to the Hausdorff measure from restricting E to the σ-field of Carathéodory measurable sets [3].

Definition 2

(Hausdorff Dimension). The Hausdorff Dimension of E is defined by

ϕ (E)

where:

H^{d} (E) = \{\begin{matrix} \infty & if 0 \leq d < ϕ (E) \\ 0 & if ϕ (E) < d < \infty \end{matrix}

(1.0.3)

1.1. Generalized Hausdorff Measure

H^{ϕ (E)} (E)

is zero or infinity, consider the following:

Definition 3

(Generalized Hausdorff Measure). Suppose

(X, d)

is a metric space. Let

h : [0, \infty) \to [0, \infty)

be an (exact) dimension function (or gauge function) which is monotonically increasing, strictly positive, and right continuous [4].

For

i \in N

, where

δ \in R

and

δ > 0

, if the Hausdorff dimension is

ϕ (E)

; we define:

H_{δ}^{h} (E) = N_{ϕ (E)} inf \{\sum_{i = 1}^{\infty} h (diam (C_{i})) : diam (C_{i}) \leq δ, E \subseteq ⋃_{i = 1}^{\infty} C_{i}\}

(1.1.1)

such if the infimum of the equation above is taken over the countable covers of sets

C_{i} \subset X

of E (which satisfy

diam (C_{i}) \leq δ

), the h-Hausdorff Outer Measure follows:

H^{h} (E) = sup_{δ > 0} H_{δ}^{h} (E) = lim_{δ \to 0} H_{δ}^{h} (E)

(1.1.2)

where when

ϕ (E) \in N

H^{h} (E)

should coincide with the

ϕ (E)

-dimensional Lebesgue Measure such that we define the "outer h-Hausdorff measure" as h-Hausdorff measure by restricting the Outer Measure to E measurable in the sense of carathèodory, where

H^{h} (E)

is strictly positive and finite.

2. Measuring "Uniformity" of a Measurable Subset of $[0, 1] \times [0, 1]$

Using this answer [5], let

S : = {[0, 1]}^{2}

be the unit square. "Partition" S naturally into four congruent squares

S_{1, j}

(with side length

1 / 2

each), where

j = 1, \dots, 4

; the quotation marks are used here because the

S_{1, j}

’s will have some common boundary points. Next, "partition" each

S_{1, j}

naturally into four congruent squares (with side length

1 / 2^{2}

each), so that we get

4^{2}

squares

S_{2, j}

for

j = 1, \dots, 4^{2}

. Continue doing so, so that at the kth step we get

4^{k}

squares

S_{k, j}

for

j = 1, \dots, 4^{k}

, for each

k = 1, 2, \dots

Take any subset A of S. For each

k = 1, 2, \dots

and each

j = 1, \dots, 4^{k}

, let

A_{k, j} : = (A \cap S_{k, j}) - s_{k, j},

where

s_{k, j}

is the southwest vertex of the square

S_{k, j}

, so that

A_{k, j} \subseteq S_{k} : = 2^{- k} S

Suppose that for each k we have a "measure"

D_{k}

of dissimilarity for subsets of

S_{k}

, so that for any two subsets B and C of

S_{k}

we have a nonnegative real number

D_{k} (B, C)

, which is the greater the more "dissimilar" B and C are (and is 0 if

B = C

); here the term "measure" is used in the general sense, not necessarily in the sense of measure theory. For instance,

D_{k} (B, C)

may depend on the Hausdorff distance [6] between B and C or on some "measure" of the symmetric difference of the sets B and C or on some combination thereof.

Then the distance of the set A from uniformity can be defined by the formula

D (A) : = \sum_{k = 1}^{\infty} \frac{1}{L^{k}} \sum_{j = 1}^{4^{k}} \sum_{m = 1}^{4^{k}} \frac{D_{k} (A_{k, j}, A_{k, m})}{1 + D_{k} (A_{k, j}, A_{k, m})},

where L is a real number

> 16

(to ensure the convergence of the series). Then

D (A)

will be small if, for "most" levels k of "zooming", "most" of the intersections of the set A with all the "k-level" small squares

S_{k, j}

"look similar" to one another. In other terms:

Definition 4

(Definition of Uniform A in The Unit Square). If

D (A) = 0

, A is uniform in

{[0, 1]}^{2}

(Of course,

D (A)

will depend on the choices of L and the dissimilarity "measures"

D_{k}

For instance, for any L and any

D_{k}

’s we have

D (S) = 0

– of course, the unit square S is at distance 0 from uniformity (in itself).

As another example, for the uniform grid

G_{n}

(defined in

§ 0, crit . (0)

) with

n = 2^{K}

for a natural K, any real

L > 16

, and any

D_{k}

’s we have

D (G_{n}) \leq \sum_{k = K + 1}^{\infty} \frac{1}{L^{k}} 16^{k} = C n^{- p} \to 0

n = 2^{K} \to \infty

, where

C : = \frac{16}{L - 16}

and

p : = {log}_{2} \frac{L}{16}

2.1. Specific Example of Measure of Uniformity for Measurable Subsets of ${[0, 1]}^{2}$

For example, if

D_{k}

H^{h}

(def. 3) where h is the dimension function of A and

L = 32

, one measure of uniformity is:

D (A) : = \sum_{k = 1}^{\infty} \frac{1}{32^{k}} \sum_{j = 1}^{4^{k}} \sum_{m = 1}^{4^{k}} \frac{H^{h} (A_{k, j} Δ A_{k, m})}{1 + H^{h} (A_{k, j} Δ A_{k, m})},

where if

H^{h} (A_{k, j} Δ A_{k, m}) = + \infty

for some

k, j, m \in N

or if

H^{h} (A_{k, j} Δ A_{k, m}) = 0

for all

k, j, m \in N

, take

§ 3

of this paper [7] (if version 3 of the paper exists, consider that version instead).

In [7], for (

§ 3

, eq. 4.1.9) if we replace k with z, such that

(F_{z}^{★ ★ ★})

is a chosen sequence from a set of equivalent ★-sequence of sets (

§ 2

, def. 4), this should give us:

\forall (ϵ > 0) \exists (N \in N) \forall (z \in N) (z \geq N \Rightarrow |D^{★} (A) - \sum_{k = 1}^{\infty} \frac{1}{32^{k}} \sum_{j = 1}^{4^{k}} \sum_{m = 1}^{4^{k}} \frac{H^{h} ((A_{k, j} Δ A_{k, m}) \cap F_{z}^{★ ★ ★})}{1 + H^{h} ((A_{k, j} Δ A_{k, m}) \cap F_{z}^{★ ★ ★})}| \leq ϵ)

where

D^{★} (A)

is the final measure of uniformity for subset of the unit square (if it exists).

References

Bonis, T. Improved rates of convergence for the multivariate Central Limit Theorem in Wasserstein distance, 2023, [arXiv:math.PR/2305.14248]. https://arxiv.org/pdf/2305.14248.pdf. 2023. [CrossRef]
(https://stats.stackexchange.com/users/346978/jaewon lee), J.L. Is there a way to measure uniformness of points in a 2D square? Cross Validated, [https://stats.stackexchange.com/q/599023]. https://stats.stackexchange.com/q/599023 (version: 2022-12-14).
M., T. The Caratheodory Construction of Measures. https://mtaylor.web.unc.edu/wp-content/uploads/sites/16915/2018/04/measch5.pdf.
Wikipedia. Dimension Function. https://en.wikipedia.org/wiki/Dimension_function.
(https://mathoverflow.net/users/36721/iosif pinelis), I.P. Defining a measure of uniformity for measurable subsets of [0,1]² w.r.t dimension α∈[0,2]. MathOverflow, [https://mathoverflow.net/q/449772]. https://mathoverflow.net/q/449772 (version: 2023-06-29).
Rockafellar, R.; Wets, R. Variational Analysis; Vol. 317, 2004; p. p.117. https://sites.math.washington.edu//~rtr/papers/rtr169-VarAnalysis-RockWets.pdf. [CrossRef]
Krishnan, B. Finding an Extension of the Expected Value That Is Unique, Finite, and Natural for All Functions in Prevalent Subset of the Set of All Functions. Preprints.org 2023. https://www.preprints.org/manuscript/202307.0560/v2. [CrossRef]

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Measuring the uniformity of Measurable Subsets of The Unit Square

Abstract

0. Intro

1. Preliminary Definitions

1.1. Generalized Hausdorff Measure

2. Measuring "Uniformity" of a Measurable Subset of [ 0 , 1 ] × [ 0 , 1 ]

2.1. Specific Example of Measure of Uniformity for Measurable Subsets of [ 0 , 1 ] 2

References

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2. Measuring "Uniformity" of a Measurable Subset of $[0, 1] \times [0, 1]$

2.1. Specific Example of Measure of Uniformity for Measurable Subsets of ${[0, 1]}^{2}$