1. Introduction

The central limit theorem in mathematical probability theory reckons that, often, the addition of independent random variables to the original variables causes the normalized sum of such variables to tend toward a normal distribution even though they are not normally distributed. However, the central theorem affords simply an asymptotic distribution, and it requires several observations to stretch into the tails of a normal distribution. Chebyshev’s Inequality is widely used in probability theory, and; it furnishes probability bounds on wide-ranging aspects of probability distributions. The central importance of the inequality is the use in limiting the distance between the mean and a random variable, Amaresh [1]. As precious as the inequality is in doing the said work, its drawback is the inability to employ it in setting confidence intervals in estimation problems. However, the propitiation of this Chebyshev’s shortcoming is the use of concentration inequalities (see [2], [3], [4]). Besides, Roos et al. [5] provided alternative tight lower and upper bounds on the tail probability under certain conditions. Such bounds were obtained as exact solutions to semi-infinite linear programs.

Chebyshev’s inequality is, fundamentally, a consequence of the measure theory, as any resulting bounding value is to be prescribed in a “measurable set”. In the light of this, Stepaniants [6] and Billingsley [7] showed that Chebyshev’s inequality can be subsumed in the theory of Lebesgue measure. Besides the qualitative use of the generic Chebyshev’s inequality in probability measures, numerous applications of Chebyshev-type inequalities furnish suitable bounds for the estimation of some class of functions. One of such celebrated functions is depicted in the abiding maximal theorem by Hardy and Littlewood [8] - a work that has engaged the attention of Flett [9], Keith [10], among numerous others found in literature. In recent times numerous Chebyshev-type inequalities are ‘crafted’ to attend to varying needs (see Teimourian and Ghazanfari [11], Nisar et al. [12])).

Another salient class of inequality is the type that is obtainable from the Chebyshev functionals of functions of bounded variation. While numerous identities that relate to the Chebyshev functional were considered by Mitrinovi ́c et al. [13,14], the quest for the bounds for the functional was done by Cerone [15,16] and Dragomir [17].

This work supplied the combined outcome of (Lebesgue) measure theory and Chebyshev’s inequality in furnishing the method of obtaining estimates (bounds) of functions and Chebyshev-type functionals. The linear combinations of independent random variables, their control, and the maximal averages over some parameters are desired for purposes of empirical risk minimization. In effect, this line of study is quite graceful, for purposes of optimization.

2. Preliminaries

Some basic tools for the development of the content of this paper are supplied in this section.

2.1. The Measure Space

Definition 1. Let be a measurable space. A set function μ is a measure on the space if:

(i) ;

(ii) 0;

(iii) Given a sequence of disjoint A₁, A₂, . . . of -sets, and if then

(1)

The triple is called a measure space.

2.1.1. Probability Space

Definition 2. Let (Ω, F) and (Ω′, F′) be two measurable spaces. The function X : Ω→S is measurable a function and

(2)

X is called a random variable if Ω′= and F′ is the Borel σ-algebra.

If Ω is any state space that is not endowed with σ-algebra, and X : Ω→Ω′ is a function that has value in a measurable space (Ω′, F′), then the smallest σ-algebra for which X is a measurable function is

(3)

The triplet is a probability space, in which:

(i) Ω denotes the sample space describing a non-empty set of all possible outcomes of some model being performed.

(ii) F denotes an σ-algebra on the sample space (Ω ∈F).

(iii) is a probability measure if it as well satisfies the countably additive property specified in 2.1 (iii).

Probability measures has been largely used in optimization problems as seen in Saito and Takahashi [18] and Yu [19].

If X₁, X₂,… is a sequence of real-valued random variables defined on , then the supremum

is measurable (see Lowther [20]).

2.2. Excerpts from Lebesgue Integral

Let be a random variable. It has a Gaussian distribution if it has a Lebesgue measurable density p on R such that

where and are the mean and variance of X.

Let be a measure space:

(i) If is an S-summable function, and S_P is an S-partition B₁, B₂,…, B_m of X, then the lower Lebesgue sum L (f , S_P) is defined by

(ii) If and is the characteristic function of D, we get (see Axler [21])

(iii) If is S-summable, then the Lebesgue space reads,

is an measurable function from to and

(iv) Lemma 1. If , then (Markov’s inequality)

(4)

Proof. Assume c > 0. Therefore

(5)

where .

2.3. Chebyshev’s Inequality from Lebesgue Integral

The generic (probability) statement of Chebyshev’s Inequality reads:

Theorem 1. (Chebyshev’s Inequality). Let X : Ω→R be a random variable on a probability space (Ω, F, P). Suppose X has a finite expected value m and finite nonzero variance σ². Then, for any real number k > 0

(6)

Note that X above is integrable.

Theorem 2. Suppose (X, S, μ) is a measure space with μ(X) = 1 and l ∈ L¹(μ), then

(7)

The proof may easily derive from the proof of Markov inequality (4) already presented here.

Compare the inequality (7) above to Markov’s inequality (4). Consider the left-hand side of the inequality (7). We see that

Let be the characteristic function of B. Then,

(8)

let . The above shows that the integral measure in (7) encodes the mean value. Therefore, ; , we recognize the integral in the rightmost expression of (7) as equal to Var(X) = σ². If P is the probability measure, one finds that the inequality (6) is equivalent to the inequality (7). Under some condition(s), the tail estimate of a given measurable function may be obtained using Chebyshev’s inequality. Let f be a non-negative measurable function. Define a set {x ∈ E: f(x) ≥ ω ≥ 0}. The property

(9)

holds. On integration, one gets Chebyshev’s inequality in the form

(10)

Suppose g is any measurable function. The representative inequalities for g may be obtained by choosing some non-negative measurable function φ(x) and applying Chebyshev’s inequality to f = φ◦g. Thus,

(11)

The required tail estimate is [21]

(12)

Still from the measure-theoretic point, let (X,Ω,μ) be a measure space and let f be an extended real-valued measurable function defined on X. Then for any real number t > 0 and 0 < p < ∞ [22],

In general, if g is an extended real-valued measurable function, nonnegative and nondecreasing, with g(t) ≠ 0 then:

Chebyshev’s inequality may be expressed in the sense of Lebesgue integration (see Stepaniants [5]) in the theorem that follows:

Theorem 3. (Generalized Chebyshev’s Inequality). Let (Y, σ, μ) be a measure space and let f be a real-valued measurable function defined on Y. Suppose μ is the Lebesgue measure and let h be a non-negative and non-decreasing real-valued measurable function on the range of f. Then, for any positive real number n and 0 < t < ∞,

(13)

Proof: For a fixed number, n, Let G_n be a set and let Denote the characteristic function of the set G_n by χG_n. By the theorem, h is non-decreasing and it is non-negative on the range of f, therefore

(14)

Apply Lebesgue integration and integrate over Y, to get

(15)

Now,

(16)

In (13) above the last inequality holds since h is nonnegative everywhere. In conclusion, we have

(17)

2.4. Equi-measurable Functions

So far, the emphasis had been on non-decreasing functions. What happens if a function is decreasing in an interval? The relationship between the above two types of functions is well articulated in the seminal and yet abiding work by Hardy and Littlewood [8].

Assume that f(x) is a positive, bounded, and measurable in (a₀, a₁); let β(y) be a measure such that f(x) ≥ y, therefore β(y) is a decreasing function of y which vanishes for sufficiently large y. The function f*(x) is defined for x ϵ [a₀, a₁] by

(18)

It is constant in some interval (i₁, i₂) if β(y) has a discontinuity, with a jump from i₁ to i₂. It is known as the rearrangement of f(x) in decreasing order. The effective upper bound of f(x) corresponds to the upper bound of f*(x) excluding in a null set, as sets of zero measure do not count in the definition of f*(x). The functions f(x) and f*(x) are equimeasurable when the measures of the sets wherein they take up values lying in a prescribed interval are equal. In the interval (a₀, a₁) if φ(x) is a function, then

(19)

whenever each integral exists. If λ ϵ [a₀, x) is a function, the functional F is such that

(20)

Define M(x, f), the maximum average of f(x) about a point x by

(21)

Lemma 2. If a function φ(x) is positive, continuous, and non-decreasing with x, and λ = λ (x) is any measurable function such that λ ϵ (a₀, x), then [8]

(22)

Note that we may, in most cases, take a_{0 = 0,} a_{1 = 1.}

Lemma 3. If M(x) is the upper bound of

(23)

Then

(24)

The proofs of Lemma 2 and Lemma 3 are available in Hardy and Littlewood [8]). The consequence of Hardy-Littlewood maximal theorem is that for a decreasing function, f*(x), that is equi-measurable with f on (0, ∞), and for a measurable set, U, with a measure, μ, if t is a positive real number, then

(25)

It was shown in Phillips [9] that inequality of the form

(26)

holds well for all Lebesgue measurable set .

3. Estimation of Bounds

The estimation of the bounds of measurable functions is of much importance in applicable phenomena. Inequality measures contribute much to near guileless estimates. Extensive energies that have so far been expended on the theory of maximal inequalities for several decades have produced a sumptuous result. As expected, such inequalities must be rigged in the Lebesgue function space if the bounds are to be sought. This section is devoted to inequality bounds that are consequences of Chebyshev’s inequality and with the underpinning of measures.

3.1. Bounds in Function Spaces

3.1.1. Probability Bound

Many a time the linear combinations of independent random variables, their control, and the maximal averages over some parameters are desired for purposes of empirical risk minimization.

Let P be a probability distribution over some set X. An ε-net for a class of subsets X is any subset such that for any qϵ H

.

The subset G approximates the probability distribution. An ε-approximation for class H is such that [23]

The following argument derives from [24].

Let X₁,..., X_n be n independent and random variables such that E[X_i] = μ and var(X_i) ≤ σ² . Let δ ∈ (0, 1) and assume that n can be factored into n = K·W where W = 8 log(1/δ) is a positive integer. For w = 1, ..., W, let denote the average over the w-th group of k variables. Then, this average takes the form

For any w = 1,...,W, it can be shown [24] that

(27)

Moreover, define as the median of {X1,..., XW}. Then

where B ∼ Bin(W, 1/4). Evidently,

(28)

Definition 3. The space of functions L ^p (Ω), p ϵ [1, ∞), for which the p-th power is Lebesgue integrable over Ω is defined by

(29)

and the space of functions L^∞ with finite essential supremum is

(30)

where

Consider the L² ball. Let be fixed and let ε > 0. A set G is called an ε-net of with respect to a distance d(·, ·) on R^d, if and for any there exists x ∈G such that d(x, z) ≤ ε. The unit L² ball of R^d is the set of vectors u that have Euclidean norm |u|₂ at most 1, usually defined by

The upper bound on the magnitude of the smallest ε-net of is given below.

Lemma 4. [24] Let ε ∈ (0, 1). Then the unit Euclidean ball has an ε-net with respect to the Euclidean distance of cardinality

Proof. Going by [24], construct an iteration of the ε-net. Choose x₁ = 0. For a given i ≥ 2, let x_i be any such that |x−x_j|₂ > ε for all j < i. If there is no such x, stop the process. Thus, an ε-net is generated. Then, the size of the ε-net has to be controlled.

The Euclidean balls centred at and with radius ε/2 are disjoint as |x−y|₂ > ε for all Besides,

(31)

where . Thus, measuring volumes, one finds

(32)

Equivalently,

(33)

Thus, the following bound holds

(34)

Let B(x, r):= {y ∈ R^d: |x −y| < r}, r > 0, denote the open ball of radius r centred at x for any x ∈ R^d. The averaging operators ψ_r on R^d for a locally integrable function f read:

(35)

On every the operators ψ_r are contractions for which Young’s inequality

(36)

holds.

For an -measurable function, f, the maximal function is of the form

(37)

with the weak

(38)

The averages ψ_rf are uniformly bounded in magnitude and in shape as r varies. From the foregoing, we state:

Proposition 1. (Hardy-Littlewood maximal inequality)

(39a)

and

(39b)

Consider the equations describing the Borel-measurable maximal averages of f in some prescribed interval:

(40)

In equations (40) above,

(41)

is the sublinear operator encoding the Hardy-Littlewood maximal operator. The proof of (41) above may be found in Keith [10].

Suppose t is a positive real number and w is an extended real-valued positive function. Let

(42)

Lemma 5. For every k ϵ (0, 1) and every t > 0:

(43)

Proof. Following Philip [9], define the distribution function w(x) by

(44)

So and we have Since the equality

(45)

holds for every t > 0, we have

(46)

The function w(x) was assumed to be nonnegative so

since w = 0 there.

Thus

(47)

A similar approach may be used to prove (i) for i = r₁, and (ii) may be proved by using the property that

(48)

The following relations on the maximal operator are often used in estimations

Lemma 6. If f ϵ L^p, then for p >1 the following inequalities

(49a)

and

(49b)

apply.

The proof to the above theorem is readily found in Flett [9] and Keith [10].

3.2. Chebyshev Functional

The bounding of the Chebyshev functional has extensive applicability in probability problems and the bounding of special functions in applied mathematics. Some theories and identities that relate to the Chebyshev functional are treated in Mitrinovi ́c et al. [13,14] while Fink [25] looked at an applicable aspect of such inequalities. Elegant and enduring work in the quest to bound the Chebyshev functional was done by Cerone [15].

As usual, let f, g: [a,b]→ be two measurable functions. The weighted Chebyshev functional W(f, g) is defined by

(50)

with the weighted integral mean given by

where and the above integral is assumed to exist. The following equivalences hold [22],

(51)

It is conventional to present the Chebyshev functional W(f, g) as Korkine’s identity. If f, g: [a,b]→ are two equi-monotonic functions, then

(52)

provided the integrals exist. If f (x) and g(y) are synchronous functions i.e. (f (x) − f (y))(g(x) −g(y)) ≥ 0, a.e. x, y ∈[a, b], the classical Chebyshev functional inequality is satisfied by

(53)

As usual, let the triple be a measure space, with μ as a countably additive and positive measure on σ with values in R ∪ {∞}.

Given a μ-measurable function , with v(x) ≥ 0 for μ, a.e. x ∈ X, define a Lebesgue space , f is μ-measurable and . Assume If f, g: X → are μ-measurable functions and f, g, f g ∈ L_v (X, σ, μ), then the Chebyshev functional that admits v(x) may be written as

(54)

where

In Cerone [15], remarkable work was done in using the above to furnish some bounds. As well, the bounds for the modulus of complex Chebyshev functionals engaged the attention of Dragomir [17]. In Grüss [26], the Chebyshev functional was evaluated as

(55)

provided are real numbers such that

The constant ¼ is assumed sharp, as there would be no smaller number in its stead. The proof of this inequality is available in Mitrinovi ́c et al. [13].

3.3. Bound for the Generalized Prime Numbers

Beurling [27] conducted an extensive and enduring study on the asymptotic distribution of prime numbers. The distribution function N(x) for the generalized prime numbers satisfies the inequality

(56)

with C > 0.

More general is the case in which the counting function of the integer satisfies

(57)

for some positive C and γ. The prime number theorem holds if γ > 3/2, else it may fail to hold. The Chebyshev prime counting estimate

(58)

is known to hold if γ >1 but it may fail if γ < 1 (see Hall [28]). It was shown (see Hall [29]) that if the distribution function N(x) is of the form

(59)

then the Chebyshev’s lower and upper bounds are satisfied by

(60)

Following Diamond [30,31]

Let some function F: [1, ∞) →R be of bounded variation on each interval [1, x], with an accompanying Borel-Stieltjes measure, dF, on [1, ∞). Define an operator T on the measures in the form

(61)

The convolution

(62)

was seen to hold well for Beurling generalized numbers (see Diamond [31] and Zhang [32]).

Let

(63)

be the convoluted sides of (59) by , where δ encodes the Dirac measure at 1, dt is the Borel-Lebesgue measure on [1,∞) and α is a positive number. Then we state, without proof, the following lemma whose proof furnishes Chebyshev’s upper and lower bounds.

Lemma 6. Suppose that (36) holds with C > 0 and γ >1. Then there exists a positive number α depending on N(x) that satisfies the conditions: (i) (ii)

The proof of the above lemma is available in Diamond [31]. As a result, the upper bound is satisfied by

(64)

where D is a number for which given a number n such that . The lower bound is satisfied by

(65)

4. Summary and Conclusions

The achievement of a near-flawless optimal estimate is always a desirable end. Boundedness is always a sine qua non to the achievement optimality. The linear combinations of independent random variables, their control, and the maximal averages over some parameters are desired for purposes of empirical risk minimization. This work supplied the (Lebesgue) measure theory and Chebyshev’s inequality synergism in providing the method of obtaining estimates (bounds) of functions and Chebyshev-type functionals which may be applicable in such risk minimizations.

Chebyshev’s inequality in the main is a consequence of the measure theory, as any resulting bounding value is embedded in a “measurable set”. It was shown in this work that Chebyshev’s inequality and Chebyshev-type inequalities can be subsumed in the theory of (Lebesgue) measures. Besides the qualitative use of the generic Chebyshev’s inequality in probability measures, numerous applications of Chebyshev-type inequalities provide appropriate bounds for the estimation of some class of functions. This reasoning calls for an in-depth study of measure theory as it acts as a catalyst to the resolution of optimization problems.

ACKNOWLEDGEMENTS

The authors are grateful to the reviewers for their useful suggestions that led to the revision of this work.