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:

09 June 2023

Posted:

12 June 2023

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Abstract

Suppose set R is a subset of [0,1] x [0,1]. We want to define a measure of uniformity of R in the unit square using dimension d in [0,2]$ of the d-dimensional Hausdorff measure. Inorder to understand uniformity, we'll give examples in section 0 where points of R are uniform in [0,1] x [0,1]. Next in section 1, we will define preliminary definitions (e.g. Hausdorff & Counting measure) to define a uniformity of measurable subsets of the unit square. Finally, in section 2 we will define a measure of uniformity between 0 and 1 w.r.t a uniform R with Hausdorff-dimension d. (In this case, the larger the measure of uniformity, the smaller the non-uniformity w.r.t to uniform R)

Keywords:

Subject: Computer Science and Mathematics - Mathematics

Intro

Suppose we define set

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

. I want to find a measure of the "uniformity" of the distribution of points in R (on set

[0, 1] \times [0, 1]

) where the larger the measure (as it approaches value one or infinity), the smaller the non-uniformity. Note inorder to understand uniformity, here are examples of R with points

uniform

[0, 1] \times [0, 1]

If R is finite, then for $n \in N$ , suppose we partition set $[0, 1] \times [0, 1]$ into $n^{2}$ squares with length n. If there exists $n \in N$ such there is only one point at the center of each square, then we say the discrete points are uniform in $[0, 1] \times [0, 1]$ .
If we have that set R has a Hausdorff dimension of 2 ( $§, def .$ ), then 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 $([x_{1}, x_{2}] \times [y_{1}, y_{2}]) \cap R$ is $(x_{2} - x_{1}) (y_{2} - y_{1})$ , then set R is uniform in $[0, 1] \times [0, 1]$ .

For R with a Hausdorff dimension between 0 and 2, it’s unclear what a uniform distribution of points in R covering

[0, 1] \times [0, 1]

looks like. (Despite this, we will try to define the following in

§

)

Note I want to find a general theory of uniformity (for distributions of points in the unit square) such that, depending on the Hausdorff dimension of the set of points (

§, def .

), these points have a measure of uniformity between zero and one or zero and infinity.

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

(Counting Measure). Suppose

| \cdot |

is the cardinality of a set. If

A \subseteq R

and μ is some measure defined on the sigma-algebra

P (R)

, then the counting measure is:

μ (A) = \{\begin{matrix} | A | & A is finite \\ + \infty & A is infinite \end{matrix}

(1)

Definition 2

(Hausdorff Measure). Let

(X, φ)

be a metric space,

d \in [0, \infty)

and

C, E

be arbitrary sets where

C, E \subseteq X

and the diameter of C is:

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_{d}

is:

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

(2)

then we define:

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

(3)

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

), then the Hausdorff Outer Measure is:

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

such for

d \in N

H^{d} (E)

coincides with the d-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 3

(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) \leq d < \infty \end{matrix}

(4)

which is needed since constant

N_{ϕ (E)}

(eq. 5) is required for the next definition.

1.1. Generalized Hausdorff Measure

H^{ϕ (E)} (E)

is zero or infinity, consider the following:

Definition 4

(Generalized Hausdorff Measure). Suppose

(X, d)

is a metric space and

C, E \subseteq X

. 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]. If the diameter of C is:

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

such that for

i \in N

, where

δ \in R

and

δ > 0

, if the Hausdorff dimension is

ϕ (E)

where Euler’s Gamma function is Γ, and

N_{ϕ (E)}

is the constant where:

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

(5)

we then 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}\}

(6)

such that 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

), then the h-Hausdorff Outer Measure follows:

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

(7)

such that for

ϕ (E) \in N

H^{h} (E)

coincides with the

ϕ (E)

-dimensional Lebesgue Measure where we define the outer h-Hausdorff measure as the h-Hausdorff measure from restricting the Outer Measure to E measurable in the sense of carathèodory, and defining h so

H^{h} (E)

is strictly positive and finite.

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

Here is the attempt to measure the uniformity of R in

[0, 1] \times [0, 1]

w.r.t a uniformR with a Hausdorff-dimension of

d \in [0, 2]

(def. 3).

Now suppose

μ (A)

is the counting measure of A (def.

)

where:

s = \{\begin{matrix} | A | & μ (A) < + \infty \\ 1 & μ (A) = + \infty \end{matrix}

For

n, j \in \{s k : k \in N\}

(where

j ≪ n

), suppose we partition set

[0, 1] \times [0, 1]

into

n^{2}

squares with length n, such that we combine these squares to form larger squares with area

1 / (j^{2})

. Furthermore, we define

i, m, r \in N \cup \{0\}

, where

x_{i + 1} - x_{i} = 1 / n

and

0 \leq i + r \leq n - 1

such that the side of each of the squares (parallel to the x-axis) is the interval

[x_{i}, x_{i + r}]

with length

1 / (n + r)

, and we define

y_{m + 1} - y_{m} = 1 / n

where

0 \leq m + r \leq n - 1

and the side of each square (parallel to the y-axis) is the interval

[y_{m}, y_{m + r}]

with length

1 / (n + r)

d \in [0, 2]

, where the d-dimensional Hausdorff measure defined on the

σ

-algebra of caratheodory-measurable sets is

H^{d}

(def. 2), I would like to divide measure

H^{d}

of each

([x_{i}, x_{i + r}] \times [y_{m}, y_{m + r}]) \cap R

(such that the area of

[x_{i}, x_{i + r}] \times [y_{m}, y_{m + r}]

1 / (j^{2})

) by the d-Hausdorff measure of all uniform R, with Hausdorff-dimension d, in all squares or

((H^{d} (dom (R)) \cdot H^{d} (range (R))) / (j^{2}))

. This results in a discrete probability distribution

P

\begin{matrix} S (n, j) = { & (i, m, r) : i, m, r \in N \cup \{0\}, x_{i + 1} - x_{i} = y_{m + 1} - y_{m} = 1 / n, \\ Area ([x_{i}, x_{i + r}] \times [y_{m}, y_{m + r}]) = 1 / j^{2}} \end{matrix}

(8)

\begin{matrix} P (n, j) = & \{\frac{H^{d} (([x_{i}, x_{i + r}] \times [y_{m}, y_{m + r}]) \cap R)}{((H^{d} (dom (R)) \cdot H^{d} (range (R))) / (j^{2})) \cdot | S (n, j) |} : (i, m, r) \in S (n, j)\} \end{matrix}

(9)

We then want to take the entropy of the discrete distribution

()

E (P (n, j)) = \sum_{x \in P (n, j)} - x {log}_{2} x

(10)

Next, we wish to define the uniformity of R in

[0, 1] \times [0, 1]

using

μ (A)

and the dimension d; however, it’s difficult (when

n, j \to \infty

and

j \leq μ (A)

such that

j ≪ n

) to compare

E (P (n, j))

with the entropy [5] of a discrete uniform distribution with the same number of elements or

{log}_{2} (| S (n, j) |)

. (Note the smaller the absolute difference of

E (P (n, j))

and

{log}_{2} (| S (n, j) |)

, the closerR is to having the same uniformity in

[0, 1] \times [0, 1]

as a uniform R, with Hausdorff-dimension d, in

{[0, 1]}^{2}

Therefore, the following might be more useful:

\bar{G} (n^{'}, j^{'}) = inf_{j \in N} inf_{n \in N} \{| P (n, j) | : n^{'}, j^{'} \in N, E (P (n, j)) \geq {log}_{2} (| S (n^{'}, j^{'}) |)\}

\underset{̲}{G} (n^{'}, j^{'}) = sup_{j \in N} inf_{n \in N} \{| P (n, j) | : n^{'}, j^{'} \in N, E (P (n, j)) \leq {log}_{2} (| S (n^{'}, j^{'}) |)\}

This gives us a measure of uniformity of R w.r.t a uniform d-dimensional space in

[0, 1] \times [0, 1]

i.e.

U^{*} (f, d, [0, 1] \times [0, 1])

(when it exists):

U^{*} (f, d, [0, 1] \times [0, 1]) = lim_{j^{'} \leq μ (A), j^{'} \to \infty} lim_{n^{'} \to \infty} \frac{\bar{G} (n^{'}, j^{'})}{| S (n^{'}, j^{'}) |} = lim_{j^{'} \leq μ (A), j^{'} \to \infty,} lim_{n^{'} \to \infty} \frac{\underset{̲}{G} (n^{'}, j^{'})}{| S (n^{'}, j^{'}) |}

Note the larger

U^{'} (f, d, [0, 1] \times [0, 1])

is, the closerR is to have the same uniformity in

[0, 1] \times [0, 1]

as a uniform R (in

{[0, 1]}^{2}

) with Hausdorff-dimension d.

2.1. Summary:

If the Hausdorff Dimension (def. 3) of set

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

ϕ (R)

and:

If $d < ϕ (R)$ then $U^{*} (f, d, [0, 1] \times [0, 1]) = 1$
If $d = ϕ (R)$ and $U^{*} (f, d, [0, 1] \times [0, 1]) \in (0, 1)$ then $U^{*} (f, d, [0, 1] \times [0, 1])$ is the final measure of uniformity; otherwise, we set dimension d to gauge function h (def. 4) or when neither of them work and R is fractal, we apply [6]
If $d > ϕ (R)$ then $U^{*} (f, d, [0, 1] \times [0, 1]) = 0$

For example, if R is a continuous function with domain

[0, 1]

(and

ϕ (R) = 1

), then if

d = 2

the measure of uniformity should be 0. (This means w.r.t a uniform R in

[0, 1] \times [0, 1]

with Hausdorff dimension 2, the previous set R is completely non-uniform.)

3. Question:

Is there a simpler measure that measures the "non-uniformity" of set R w.r.t a uniform R that has a dimension

d \in [0, 2]

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/abs/2305.14248.
(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.
M., G. 2 ed.; Springer New York: New York [America];, 2011; pp. 61–95. https://ee.stanford.edu/~gray/it.pdf, doi:https://doi.org/10.1007/978-1-4419-7970-4. [CrossRef]
B., B.; A., F. Ratio Geometry, Rigidity And The Scenery Process For Hyperbolic Cantor Sets. https://arxiv.org/pdf/math/9405217.pdf.

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Measuring the uniformity of Measurable Subsets of The Unit Square

Abstract

Intro

1. Preliminary Definitions

1.1. Generalized Hausdorff Measure

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

2.1. Summary:

3. Question:

References

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