International Journal of Probability and Statistics

p-ISSN: 2168-4871    e-ISSN: 2168-4863

2022;  11(1): 1-8


Received: Dec. 16, 2021; Accepted: Jan. 12, 2022; Published: Jan. 21, 2022


Lebesgue−Chebyshev Synergism for the Bounds of Measurable Random Variables

Francis Egenti Nzerem, Ukamaka Cynthia Orumie

Department of Mathematics and Statistics, University of Port Harcourt, Choba, Nigeria

Correspondence to: Francis Egenti Nzerem, Department of Mathematics and Statistics, University of Port Harcourt, Choba, Nigeria.


Copyright © 2022 The Author(s). Published by Scientific & Academic Publishing.

This work is licensed under the Creative Commons Attribution International License (CC BY).


The attainment of a near-faultless optimal estimate is always the chief goal in the application of measures. Boundedness is always an ‘essential commodity’ if optimality is desired. A function is Lebesgue integrable if theLebesgue integral does not ‘explode’; as well the existence of Chebyshev bound assures that the estimates of certain random variables or those of certain functionals, as the case may be, do not ‘explode’. The realization of the latter has the underpinning of Lebesgue measure applied on the set functions under consideration. Therefore, this work treated a veritable way of estimating the (maximal and minimal) bounds of Chebyshev-type inequalities, with the functions presupposed Lebesgue measurable.

Keywords: Lebesgue−Chebyshev, Inequalities, Bounds, Maximal function, Probability

Cite this paper: Francis Egenti Nzerem, Ukamaka Cynthia Orumie, Lebesgue−Chebyshev Synergism for the Bounds of Measurable Random Variables, International Journal of Probability and Statistics , Vol. 11 No. 1, 2022, pp. 1-8. doi: 10.5923/j.ijps.20221101.01.

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
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
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
The triplet is a probability space, in which:
(i) 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 SP is an S-partition B1, B2,…, Bm of X, then the lower Lebesgue sum L (f , SP) 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)
Proof. Assume c > 0. Therefore
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
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
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,
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
holds. On integration, one gets Chebyshev’s inequality in the form
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,
The required tail estimate is [21]
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 < ∞,
Proof: For a fixed number, n, Let Gn be a set and let Denote the characteristic function of the set Gn by χGn. By the theorem, h is non-decreasing and it is non-negative on the range of f, therefore
Apply Lebesgue integration and integrate over Y, to get
In (13) above the last inequality holds since h is nonnegative everywhere. In conclusion, we have

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
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
whenever each integral exists. If λ ϵ [a0, x) is a function, the functional F is such that
Define M(x, f), the maximum average of f(x) about a point x by
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]
Note that we may, in most cases, take a0 = 0, a1 = 1.
Lemma 3. If M(x) is the upper bound of
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
It was shown in Phillips [9] that inequality of the form
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,..., Xn be n independent and random variables such that E[Xi] = μ and var(Xi) ≤ 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
Moreover, define as the median of {X1,..., XW}. Then
where B ∼ Bin(W, 1/4). Evidently,
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
and the space of functions L with finite essential supremum is
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 Rd, if and for any there exists x G such that d(x, z) ≤ ε. The unit L2 ball of Rd 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 xi be any such that |x−xj|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,
where . Thus, measuring volumes, one finds
Thus, the following bound holds
Let B(x, r):= {yRd: |x −y| < r}, r > 0, denote the open ball of radius r centred at x for any xRd. The averaging operators ψr on Rd for a locally integrable function f read:
On every the operators ψr are contractions for which Young’s inequality
For an -measurable function, f, the maximal function is of the form
with the weak
The averages ψrf are uniformly bounded in magnitude and in shape as r varies. From the foregoing, we state:
Proposition 1. (Hardy-Littlewood maximal inequality)
Consider the equations describing the Borel-measurable maximal averages of f in some prescribed interval:
In equations (40) above,
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
Lemma 5. For every k ϵ (0, 1) and every t > 0:
Proof. Following Philip [9], define the distribution function w(x) by
So and we have Since the equality
holds for every t > 0, we have
The function w(x) was assumed to be nonnegative so
since w = 0 there.
A similar approach may be used to prove (i) for i = r1, and (ii) may be proved by using the property that
The following relations on the maximal operator are often used in estimations
Lemma 6. If f ϵ Lp, then for p >1 the following inequalities
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
with the weighted integral mean given by
where and the above integral is assumed to exist. The following equivalences hold [22],
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
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
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. xX, define a Lebesgue space , f is μ-measurable and . Assume If f, g: X → are μ-measurable functions and f, g, f g Lv (X, σ, μ), then the Chebyshev functional that admits v(x) may be written as
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
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
with C > 0.
More general is the case in which the counting function of the integer satisfies
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
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
then the Chebyshev’s lower and upper bounds are satisfied by
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
The convolution
was seen to hold well for Beurling generalized numbers (see Diamond [31] and Zhang [32]).
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
where D is a number for which given a number n such that . The lower bound is satisfied by

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.


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


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