A Converse of Tchebyshev Inequality

Looking at Harnack inequality in Harmonic Spaces, we complete the well-known Tchebyshev inequality for positive integrable function with an opposite one.


Introduction
The purpose of this paper is to give a counter part for the famous Tchebyshev inequality where f is a positive measurable function and α is a strictly positive number. Namely, if in addition, μ α < fdμ, then we have an opposite inequality In the second part, we give a proof of Kolmogorov theorem on the existence of an infinite product of probability measure using the above inequality and Daniell procedure.

The Main Result
In this section, we consider a measurable space (X, A), where X is an arbitrary nonempty set and A is a σ-algebra of subsets of X. For any positive real measure μ on A, we denote by μ the total mass of μ, i.e., μ = μ(X) and for any function f : X → R and any real number α, we put Particularly, if μ is a probability measure, we have Proof. We begin with the following special case, where P is a probability on A and the function g is A-measurable, 0 ≤ g(x) ≤ 1 for all points x ∈ X.
The following relations are obvious. The function 1 − g being positive we can apply the classical Tchebyshev inequality, namely Using now the relations (1) and (2), we get The general case, when f is a positive, bounded A-measurable function on X and μ is a positive real measure reduces to the above special case considering the function g on X and the probability P on A given by Taking now a positive real number α such that α μ < fdμ, we have α < fdP and, therefore,

A Converse of Tchebyshev Inequality
Page 3 of 8 202 From the above special case, we get On the other hand, we have g > α f = [f > α] and, therefore, from relation (4), we get: The last part of Theorem follows from (5) taking into account that μ = 1, when μ is a probability on A.
Particularly, there exists x 0 ∈ X such that f n (x 0 ) ≥ β for all n ∈ N.
Proof. Indeed, using the Lebesgue convergence theorem, we have Obviously, inf n f n ≤ inf n f n and applying the opposite Tchebyshev inequality to the function inf f n , we get and therefore, the proof is finished taking into account that

Application to Kolmogorov-Ionescu Tulcea-Neveu Theorem
We consider a family (Ω i , K i , P i ) i∈I of probability spaces. If F(I) = F is the set of all finite subsets of I, then for any F ∈ F, we consider, as usually, the finite product of probability spaces (Ω F , is the σ-algebra on Ω F generated by the sets × i∈F A i , where A i ∈ K i for any i ∈ F and P F is the unique probability on K F such that 202 Page 4 of 8 Also for any K F -measurable, real, bounded function f on Ω F , we denote by L F (f ) the real number fdP F . Using Fubini Theorem, [4,5] and regarding f as a function on the space Further, we denote by (Ω I , K I ) the measurable space, where Ω I = × i∈I Ω i and K I is the smallest σ-algebra on Ω I containing the "cylinder" where F runs F. Obviously, if for any F ∈ F and any M ∈ K F , we denotẽ M := M × × j∈I\F Ω j , thenM ∈ K I and the σ-algebraK F given bỹ is a σ-algebra included in K I , isomorphic with K F , i.e., the map M →M from K F intoK F is a bijection preserving the order relation: If π F is the projection of Ω I on Ω F , thenK F = π −1 F (K F ) and a function h : Ω F may be identified with a functionh on Ω I depending on the variable Also the measure P F on (Ω F , K F ), F ∈ F(I) may be identified with the measureP F onK F , namelỹ This is why we shall write K F , P F , h instead ofK F ,P F ,h . . . With these conventions, on the measurable product space (Ω I , K I ), we have a family (K F ) F ∈F (I) of σ-subalgebras of K I , K F ⊂ K F if F ⊂ F and a family (P F ) F of product probabilities P F : K F → [0, 1], namely Moreover, for any real bounded K F -measurable function f on Ω I , we have and the Fubini theorem may be applied for such function.
Further, we denote by D the set of all bounded real functions f on Ω which is B F f -measurable for some F f ∈ F(I). Obviously, D is a real vector lattice of functions on Ω which contains the constant functions and we have If we denote by L(f ) the number fdP F , then obviously L : D → R is an increasing linear functional such that L (1 ΩI ) = 1. With the above notations, we have Proof. Let us suppose that for the above sequence (f n ) n , we have inf n L(f n ) = β and β > 0.
It is no loss of generality if we suppose that f 1 = 1. If we denote by F n the finite subset of I such that the function f n is a K Fn -measurable, then the set n≥1 F n is countable. To simplify, we shall write Let us consider α ∈ R, 0 < α < β and let (β n ) n be a decreasing sequence of real numbers such that α < β n < β n−1 < β 1 < β. Since for any n ∈ N we have, applying Fubini theorem, and since the sequence g n : Ω 1 → R + of K 1 -measurable functions given by is decreasing and inf n g n (x 1 )dP 1 (x 1 ) ≥ β then applying Proposition 2.1, there exists a point x • 1 ∈ Ω 1 such that g n (x • 1 ) ≥ β 1 for any n ∈ N, i.e., Replacing now the starting sequence (f n ) n by the sequence Ω I y → f n (x • 1 , y) and the number β by the number β 1 , we may choose an element As a corollary of the previous theorem, we obtain Theorem 3.2. (Kolmogorov-Ionescu Tulcea-Neveu Theorem) Let (Ω i , K i , P i ) i∈I be a family of probability spaces. Then, on the product of measurable spaces (Ω I , K I ), there exists a uniquely determined probability P I such that for any finite product A = A i1 × A i2 × · · · × A i k with the A ij ∈ K ij for j ≤ k, we have Proof. With the notations from Theorem 3.1 and using Daniell Theorem [1][2][3], there exists a unique measure P on the σ-algebra making measurable all functions from the vector lattice D and such that L(f ) = fdP, ∀f ∈ D.
But the σ-algebra generated by D is exactly the σ-algebra product K I . Particularly, if f ∈ D is the characteristic function of the set π −1 F (A) A = A i1 × A i2 × · · · × A i k , F = {i 1 , i 2 , . . . , i k } , we have