Correlation Inequalities in Function Spaces

We give a condition for a Borel measure on R which is sufficient for the validity of an AD-type correlation inequality in the function space In [1] was proved that if φ1, φ2, φ3, φ4 are bounded real non negative measurable functions on the space with measure (Rn,B, μ) which satisfy for all x̄, ȳ ∈ R the following inequality φ1(x̄)φ2(ȳ) ≤ φ3(x̄ ∨ ȳ)φ4(x̄ ∧ ȳ) a.s., (1) then ∫ φ1(x̄)μ(dx̄) ∫ φ2(x̄)μ(dx̄) ≤ ∫ φ3(x̄)μ(dx̄) ∫ φ4(x̄)μ(dx̄), (2) where μ(dx̄) is the product σ−finite measure on B, (x̄∨ ȳ)i = xi∨yi, (x̄∧ ȳ)i = xi ∧ yi. That proof was simplified in [2] via induction on dimension n suggested in [3], [7] ,[8]. The question we consider here is how the problem can be viewed in the case of not arbitrary measure ν on R , when possibly T = [0, 1]? The next theorem answers this question. Let for arbitrary real functions x(t), y(t), t ∈ T =[0, 1], (x y)(t)=x(t) y(t) and (x ∧ y)(t) = x(t) ∧ y(t). Let also νi(dx̄), i = 1, 2, 3, 4 be measures on the Borel sets B(C) of the linear space C of continuous functions from R with the norm ||·||∞, which are finite on the compact subsets of C. Let also φi, i = 1, 2, 3, 4 be four uniformly bounded nonnegative Borel real functions from R . Theorem 1. If the following conditions are valid φ1(f)φ2(g) ≤ φ3(f ∨ g)φ4(f ∧ g) (3) ν1(A)ν2(B) ≤ ν3(A ∨ B)ν4(A ∧ B), A,B ∈ B(C), (4) then ∫ φ1(x̄)ν1(dx̄) ∫ φ2(x̄)ν2(dx̄) ≤ ∫ φ3(x̄)ν3(dx̄) ∫ φ4(x̄)ν4(dx̄). (5) Here A ∨ B = {a b : a ∈ A, b ∈ B}, AB = {a b : a ∈ A, b ∈ B}. Condition (4) is also necessary for (5). Indeed indicator functions IA, IB, IA B, IA ∧ B satisfy (3) and substitution of them in (5) gives (4). 1 This work is partially supported by RFFI grants No 03-01-00592 and 03-01-00098 and INTAS grant No 00-738. R. Ahlswede et al. (Eds.): Information Transfer and Combinatorics, LNCS 4123, pp. 572–577, 2006. c © Springer-Verlag Berlin Heidelberg 2006 Correlation Inequalities in Function Spaces 573 We call a measure ν which satisfies the relation ν(A)ν(B) ≤ ν(A ∨ B)ν(A ∧ B), A,B ∈ B(C) an FKG measure. Proof. If ∫ φ3(x̄)ν3(dx̄) ∫ φ4(x̄)ν4(dx̄) = ∞ then (5) follows. Next we consider that ∫ φ3(x̄)ν3(dx̄) ∫ φ4(x̄)ν4(dx̄) < ∞ and propose at first that ∫ φ1(x̄)ν1(dx̄) ∫ φ2(x̄)ν2(dx̄) < ∞. (6) Then ∫ φi(x̄)νi(dx̄), i = 1, 2, 3, 4 are finite Borel measures which are regular and hence there exists a compact set K ⊂ C, such that for given > 0 ∣∣∣∣ ∫ K φi(x̄)νi(dx̄)− ∫ φi(x̄)νi(dx̄) ∣∣∣∣ ≤ , i = 1, 2, 3, 4 (7) and ν(K) < ∞. This compact set is by Ascoli’s Lemma the set of equicontinuous functions {xt} which for some N > 0 satisfy the relation |xt| ≤ N. Without loss of generality we will consider that K is the set of all such functions. It is easy to see that this set is a distributive lattice. Indeed if |xi(t)− y(t)| < , |xi| ≤ N, i = 1, 2 then |(x1 ∨ x2)(t) − y(t)| < , |(x1 ∧ x2)(t)− y(t)| < , (8) |(x1 ∨ x2)(t)| < N, |(x1 ∧ x2)(t)| ≤ N. We consider the partition of the interval T into m consecutive subintervals Δi = [ti−1, ti), i = 1, 2, . . .m − 1, t0 = 0, Δm = [tm−1, 1] of equal length choosing m in such a way that if t, t′ ∈ Δi, then |xt − xt′ | < δ/2. (9) Without loss of generality we assume thatN is integer and that δ = L−1 for some natural L. Next we divide the interval [−N,N ] into 2N/δ = 2LN consecutive subintervals Γj = [sj−1, sj), j = 1, 2, . . . , 2LN − 1, Γ2LN = [s2LN−1, 2LN ] of equal length δ. At last we consider the partition of the compact K into the set of cylinders (m = 2LN + 1) πt0,t1,...,tm(i0, i1, . . . , im) = {xt : xtj ∈ Γij}, ij = 1, 2, . . . , 2LN. 574 R. Ahlswede and V. Blinovsky Consider the finite set of rectangles K(i0, i1, . . . , im) Δ = Kt0,t1,...,tm(i0, i1, . . . , im) = {xt :xtj ∈ Γij ; |xt−yj |<δ, t∈Δj}⊂R , where yj is the center of the interval Γj . Then from (9) it follows that K ⊂ ⋃ ij Kt0,t1,...,tm(i0, i1, . . . , im). Note also that diam(K(i0, i1, . . . , im)) = 2δ. (10) Now we approximate in L(R , νi) functions φi on the compact K by continuous functions fi on K : ∫ K |φi(x̄)− fi(x̄)|νi(dx̄) < . (11) Using a standard procedure we can choose fi in such a way that fi ≤ φi, i = 1, 2; fi ≥ φi, i = 3, 4. Note, that functions fi are uniformly continuous onK and consequently, choosing δ sufficiently small, we can choose new functions ξi, i = 1, 2, 3, 4 on K such that 0 ≤ fi − ξi < /νi(K), i = 1, 2; 0 ≤ ξi − fi < /νi(K), i = 3, 4 and every ξi is constant on every set Kt0,t1,...,tm(i0, i1, . . . , im) K. At last note that the family A of sets K(i0, i1, . . . , im) ⋂ K , ij = 1, 2, . . . , 2LN is a distributive lattice under the operations ∨ , ∧ on the set of indices ij : K(i0 ∨ i0, i1 ∨ i1, . . . , im ∨ im) ⋂ K, K(i0 ∧ i0, i1 ∧ i1, . . . , im ∧ im) ⋂ K ∈ A. Hence we have eight families of values νi(i0, i1, . . . , im) Δ = νi(K(i0, i1, . . . , im)), i = 1, 2, 3, 4, ξi(i0, i1, . . . , im) Δ = ξi(x̄), x̄ ∈ K(i0, i1, . . . , im)