Weak Convergence of Convolution Products of Probability Measures on Semihypergroups

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1 Int. Journal of Math. Analysis, Vol. 8, 2014, no. 3, HIKARI Ltd, Weak Convergence of Convolution Products of Probability Measures on Semihypergroups Norbert Youmbi Department Of Mathematics Saint Francis University Loretto, PA USA Copyright c 2014 Norbert Youmbi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Let S be a topological semihypergroup. As it is known for hypergroups, the lack of an algebraic structure on a semihypergroup pause a serious challenge in extending results from semigroups. We use the notion of concretization or pseudomultiplication, to prove some results on weak convergence of the sequence of averages of convolution powers of probability measures on topological semihypergroups. As an application we provide an alternative method of solving the Choquet Equation on hypergroups. Mathematics Subject Classification: 43A62 Keywords: Weak convergence; Choquet Equation; Semihypergroups; Hypergroups; Concretization 1 Introduction A topological semihypergroup S is defined by dropping the requirement of an involution or an identity element from the definition of a hypergroup. Results from topological semigroups could easily be extended to semihypergroups, but some present serious challenges due to the fact that a semihypergroup like a hypergroup does not have a direct algebraic structure. In most cases we use the algebraic structure inherited from the measure algebra space M(S). In

2 110 Norbert Youmbi this paper we will use the notion of concretization or pseudomultiplication to prove results on the sequences of averages of convolution powers of probability measures. Concretization was used in the case of one dimensional hypergroups by Zeuner [Ze89]. We use the same definition with necessary adjustments for semihypergroups. Our results were first considered in the case of topological semigroups by Högnäs G. and Mukherjea A in [HM95]. All undefined terms used in this work in connection with topological semihypergroups can be found in Jewett [Je75] or Youmbi [Yo12]. We start with some standard basic definitions and notations. Let S be a locally compact Hausdorff space : C(S): the space of complex continuous functions on S; C b (S): the space of bounded elements of C(S); C 0 (S): the space of elements of C b (S) which tends to 0 at ; C c (S): the space of elements of C 0 (S) with compact support; C c + (S): the space of nonnegative elements of C c (S); M(S) denotes the set of finite regular Borel measures; M + (S) the set of non-negative measures; M 1 (S) denote the set of probability measures; If μ M(S) then Supp(μ) ={x S :ifv is any open set containing x then μ(v ) > 0}; an unspecified topology on M + (S) is the cone topology. Definition 1.1 A nonempty locally compact Hausdorff space S will be called a semihypergroup if the following conditions are satisfied: (SH 1 ) (M(S), +, ) is a Banach algebra. (SH 2 ) For all x, y S, δ x δ y is a probability measure with compact support. (SH 3 ) The mapping (x, y) δ x δ y of S S into M 1 (S), where S S has the product topology and M 1 (S) has the weak topology, is continuous. (SH 4 ) The mapping(x, y) Supp(δ x δ y ) of S S into C(S) is continuous, where C(S) is the space of compact subsets of S endowed with the Michael topology, that is the topology generated by the subbasis of all C U (V )= {C C(S) :C U and C V } where U and V are open subsets of S. If in addition, SH 5 SH 6 there exists e S such that δ x δ e = δ e δ x = δ x x S, and There exists a topological involution (a homeomorphism) from S onto S such that (x ) = x x S, with (δ x δ y ) = δ y δ x and e Supp(δ x δ y ) if and only if x = y where for any Borel set B, μ (B) = μ({x : x B}). (S, ) will be called a hypergroup. Remark 1.1 (i) If δ x δ y = δ y δ x for all x, y S we say that (S, ) is a commutative semihypergroup.

3 Weak convergence of convolution products 111 (ii) The convolution on M(S) is defined by μ ν(f) = fdμ ν = μ(dx) for all f C b (S). S S S ν(dy) fdδ x δ y. S Example 1.1 define 1. Let S = {e, a, b}. Lete be the identity element and let us δ a δ a = 1 2 δ a δ b δ b δ b = δ a δ a δ b = δ b δ a = 1 2 δ e δ b Then (S, ) is a semihypergroup and if we defined an involution by a = b and b = a we have (δ a δ a ) = 1 2 δ a δ b = 1 2 δ b δ a But δ a δ a = δ b δ b = δ a 1 2 δ a δ b, although e Supp(δ a δ b ) this involution does not satisfy the condition (δ a δ b ) = δ b δ a this semihypergroup is almost (though not) a hypergroup and it is called a regular semihypergroup. 2. Let H = {e, x, y} and let e be the identity element, the identity function is considered as the involution, and a commutative convolution is defined on H by δ x δ x = aδ e + bδ x + cδ y δ y δ y = a δ e + c δ x + b δ y δ x δ y = δ y δ x = qδ x + q δ y Then (H, ) is a hypergroup provided a + b + c = a + b + c = q + q =1 (for the convolution of two point masses to be a probability measure, and a c = aq (for associativity of convolution). Definition An element e S is called a left (right) identity element of S if δ e δ x = δ x ( δ x δ e = δ x ) for every x S. An element e is called a two sided identity of S or simply an identity of S, if it is both a left and right identity. The identity, when it exists, is unique. 2. An element z S is called a left(right) zero element of S if δ z δ x = δ z (δ x δ z = δ z ) for all x S. Ifz is both left and right zero, we simply call it the zero of S. A semihypergroup has at most one zero.

4 112 Norbert Youmbi 3. An element a S is called an idempotent element of S if δ a δ a = δ a Remark 1.2 The only idempotent element in a hypergroup is the identity element. For if there is an idempotent element, its point mass would be an idempotent measure and its support a singleton Definition 1.3 Let S be a locally compact semihypergroup. The center of S is defined by Z(S) ={x S : Supp(δ x δ y ) is a singleton, for all y S} Remark 1.3 For a hypergroup H the center is the maximum subgroup defined by Jewett as Z(H) ={x H : δ x δ x = δ x δ x = δ e }. Example 1.2 i. Every semigroup is a semihypergroup and its center is the entire semigroup. Also every group is a hypergroup which is the maximum subgroup( equivalently the center) of itself. ii. If H is a hypergroup, then e H so the center of a hypergroup is nonempty. When Z(H) ={e}, the center is said to be trivial. iii. Let S = {x, y} with convolution defined by δ x δ x = δ y δ y δ y = 1 4 δ x δ y δ x δ y = δ y δ x = 1 2 δ x δ y from the definition of two-elements semihypergroups above Example??, S is a semihypergroup with a void center. iv. Consider the segment [0, 1] with convolution defined by δ r δ s = 1 2 δ r s δ 1 1 r s for all r, s [0, 1] Zeuner [Ze89] proved that ([0, 1], ) is a hypergroup with a nontrivial center {0, 1}. Definition A subsemihypergroup L (R) of a semihypergroup S is called a left (right) ideal of S if S L L (R S R); I is called an ideal of S if and only if it is both a right and left ideal. 2. S is called, left (right) simple if it contains no proper left (right) ideal. S is said to be simple if it contains no proper ideal. A left (right) ideal is said to be a principal left (right) ideal if it is of the form {a} Sa ( {a} as)for some a S (Recall that we write Sa to mean S {a}).

5 Weak convergence of convolution products a, b S we say that the equation xa = b is solvable if and only if there exists x 0 S such that b Supp(δ x0 δ a ) Definition An idempotent element in a semihypergroup S is said to be a primitive idempotent element if it is in the center of the semihypergroup and is minimal with respect to the partial order on E(S) (the set of idempotent elements of S), defined by e f δ e δ f = δ f δ e = δ e 2. A completely simple semihypergroup is a simple semihypergroup which contains a primitive idempotent element. Remark 1.4 The order defined on E(S) uses convolution of point masses to compare idempotent elements of S. Note that if a is a primitive idempotent of S, δ a is not necessarily a primitive idempotent in M 1 (S), according to the definition of primitive idempotents in the semigroup (with respect to convolution)m 1 (S). Definition 1.6 A completely simple minimal two-sided ideal of a semihypergroup is called its kernel. From now on, S will denote a locally compact Hausdorff second-countable semihypergroup. (Some of the results are valid in more general topological structures; however this is not often pointed out explicitly). We recall that (from Banach-Alaoglu s theorem in functional analysis that the unit ball in the dual of C c (S) is weak* compact) the set B(S) {μ : μ M(S) + with μ(s) 1} is compact in the weak* topology. Recall: A net (μ α )inb(s), w converges to μ in B(S) if and only if for every f in C c (S), fdμ α fdμ. However, P (S) {μ B(S) :μ(s) =1} need not be weak* compact, unless S is compact. Note that in P (S), weak* compactness is equivalent to weak compactness, and thus P (S) is weak* compact if and only if S is compact. For a subset Γ P (S), the weak* closure of Γ in P (S) is weak* compact, if Γ is tight; that is, given ɛ>0, there is a compact subset K ɛ S such that μ Γ μ(k ɛ ) > 1 ɛ The reason for this is obvious since μ w -closure of Γ and Γ is tight only if μ P (S) and since B(S) isw -compact. Definition 1.7 If f is a Borel function on S and x, y S, then we define f(x y) f x (y) f y (x) = fd(δ x δ y ) If this integral exists, even when it is not finite, f x is called the left translation of f and f x is called the right translation of f. S

6 114 Norbert Youmbi The next two lemmas are proved in [Je75] Lemma 1.1 Let f be a continuous function on S and let x S i. The mapping (x, y) f(x y) is a continuous function on S S ii. f x and f x are continuous functions on S. Lemma 1.2 Let f B (S), μ, ν M + (S) and x, y, z S i. The mapping (x, y) f(x y) is a Borel function on S S ii. f x and f x are Borel functions in S iii. S fd(μ ν) = S iv. f S xdμ = fd(δ S x μ) v. f x (y z) =f z (x y) S f(x y)μ(dx)ν(dy) Notation 1.1 Let S be a locally compact semihypergroup. Then x S, μ M 1 (S), and f C(S), we write: δ x μ(f) = f x dμ ( μ(f x ), say) S and also, μ δ x (f) =μ(f x ) Definition 1.8 Let S be a locally compact semihypergroup and B be a Borel subset of S. Then Bx = {y S : Supp(δ y δ x ) B } Similarly, x B = {y S : Supp(δ x δ y ) B } In the next section we introduce the notion of concretization for semihypergroup. Most results are simple generalization of results on concretization as defined by Zeuner [Ze89] for hypergroups.

7 Weak convergence of convolution products Concretization for Semihypergroups Definition 2.1 A triplet (X, μ, Φ) consisting of a compact space X, a probability measure μ M 1 (X), and a Borel-measurable mapping Φ:S S X S is called a concretization of the semihypergroup (S, ) if For all x, y S and A B(S). μ({z X : φ(x, y, z) A}) =δ x δ y (A) Example Let G be a locally compact group with multiplication, a convolution and a neutral element e. The triplet (X, μ, Φ) defined by X = {e}, μ = δ e and Φ(x, y, e) :=xy for all x, y G is a concretization of G. 2. Consider the hypergroup K = R + with convolution defined by δ x δ y = 1 2 δ x y δ x+y for all x, y K we obtain the concretization (X, μ, Φ) where X = { 1, 1},μ= 1 2 δ δ 1 and Φ(x, y, 1) = x y Φ(x, y, 1) = x + y Since Φ is Borel measurable we just need to check that μ({z X :Φ(x, y, z) A}) =δ x δ y (A) Actually μ({z X : φ(x, y, z) A}) = δ x δ y (A) = 1 2 δ x y (A)+ 1 2 δ x+y(a) And since X = { 1, 1}, then if Φ(x, y, z) / A z { 1, 1}, then x y / A and x + y / A so that μ({z X :Φ(x, y, z) A}) =0 and δ x δ y (A) = 0. If Φ(x, y, 1) A and Φ(x, y, 1) / A, then μ({ 1}) = 1 and δ 2 x δ y (A) = 1δ 2 x y (A) = 1 and if Φ(x, y, 1) / A 2 and Φ(x, y, 1) A then μ({1}) = 1 and δ 2 x δ y (A) = 1 δ 2 x+y(a) = 1. 2 Finally if Φ(x, y, 1) A and Φ(x, y, 1) A then μ({ 1, 1}) =1and δ x δ y (A) =1. So we have (X, μ, Φ) as defined above is a concretization of (R +, ).

8 116 Norbert Youmbi The next theorem is from [BH95] it is also valid for semihypergroups with the same proof. Theorem 2.1 Let S be a second countable semihypergroup. There exists a measurable mapping Φ from S S [0, 1] into S such that ([0, 1],λ [0,1], Φ) is a concretization of S. Remark 2.1 In the special case of one dimensional semihypergroup S = R + we may assume without loss of generality that minsupp(δ x δ y )= x y maxsupp(δ x δ y )=x + y whenever x, y K The measurable mapping Φ:S S [0, 1] S established in Theorem 2.1 also satisfies the following five properties: 1. Φ(x, y, 0) = x y 2. Φ(x, y, 1) = x + y 3. Φ(x, y, t) =Φ(y, x, t) t [0, 1] 4. Φ(0,x,t)=Φ(x, 0,t)=x t ]0, 1] 5. The mapping Φ(.,., t) :S S S is lower semicontinuous. Now let S be a semihypergroup with a fixed concretization (X, μ, Φ) and (Ω, A, P) denote an arbitrary probability space. Definition 2.2 For any S-valued random variables X and Y on (X n ) n 1 and an (auxiliary) X-valued random variable ξ on (Ω, A,P) such that ξ is (stochastically) independent of X Y and has distribution P ξ = μ we define the randomized sum of X with Y by X ˆ+Y =Φ(X, Y, ξ). Remark 2.2 This definition can be extended to sequences (X n ) n 1 of X- valued random variables on (Ω, A,P) provided all random variables occurring in the sequence (X n ) n 1 and (ξ n ) n 1 are independent and P ξn := μ for all n 1 in fact by the recurrence 0 ˆX j := e n j=1 j=1 n 1 ˆX j := X n ˆ+ ˆX j,n 1 j=1

9 Weak convergence of convolution products 117 the randomized sums S n = n j=1 ˆX j, n 1 are introduced again as S-valued random variables on (Ω, A,P), which form a (non homogeneous) Markov chain (S n ) n 0 with corresponding sequence (N n ) n 1 of transition kernels on (S, B(S)) satisfying N n (x, A) =(P Xn δ x )(A) = P (S n A : S n 1 = x) For P Sn 1 -almost all x S, A B(S) and n 1 Proposition 2.1 Let X and Y be S-valued random variables and let ξ be an X-valued random variable on (Ω, A,P) with P ξn := μ such that X, Y, ξ are independent then P X ˆ+Y = P X P Y Proof: A (S, B(S)) So P X ˆ+Y = P X P Y P X ˆ+Y (A) =P (Φ(X, Y, ξ) A) = P (Φ(X, Y, ξ) A)P X (dx)p Y (dy) μ[φ(x, Y, ξ) A)]P X (dx)p Y (dy) δ x δ y (A)P X (dx)p Y (dy) P X P Y (A) Remark 2.3 Forming randomized sums is generally not an associative operation although convolution obviously is. While randomized sum X ˆ+Y clearly depends on the particular choice of the underlying concretization of S the joint distribution of the random variables X, Y and X ˆ+Y does not. 3 Sequence of Convolution Powers of Probability Measures Theorem 3.1 Let S be a locally compact semihypergroup. Assume μ M 1 (S) and suppose that the sequence (μ n ) is tight. Suppose also that S = [ Supp(μ) n ] n=1

10 118 Norbert Youmbi let K = {μ M 1 (S) :μ is a weak limit point of the sequence (μ) n } also let us define S 0 = {Supp(λ) :λ K} and S 1 = S 0 then the sequence 1 n n μ k k=1 converges weakly to a probability measure ν such that ν = ν ν = μ ν = ν μ and Supp(ν) is the closed minimal ideal of S. Proof: Write μ n = 1 n n k=1 μk then for k 1, μ k μ n = μ n μ k and lim n μ n μ k μ n = 0 (1) its follows, since the sequence μ n is tight, that the sequence (μ n ) is also tight so that {(μ n ):n 1} is weakly relatively compact. Let ν 1 and ν 2 be two limit points of (μ n ) then by ( 1) μ k ν 1 = ν 1 μ k = ν 1 It follows that and That is 1 n 1 n μ k ν 2 = ν 2 μ k = ν 2 n μ k ν 1 = 1 n k=1 n μ k ν 2 = 1 n k=1 n ν 1 μ k = ν 1 k=1 n ν 2 μ k = ν 2 k=1 μ n ν 1 = ν 1 μ n = ν 1 μ n ν 2 = ν 2 μ n = ν 2 which then implies that ν 1 = ν 2 ( ν) and μ ν = ν μ = ν = ν ν and since ν is an idempotent measure it is a simple semihypergroup and since Supp(μ) Supp(ν) = Supp(ν) Supp(μ) = Supp(ν),Supp(ν) is the minimal ideal of S = [ n=1 Supp(μ)n ] Remark 3.1 If μ n converges weakly then liminf(supp(μ) n ) is nonempty. To see this suppose μ n w ν then claim Supp(ν) liminf(supp(μ) n ) for let x Supp(ν) then for every neighborhood U of x, ν(u) > 0 but ν(u) liminfμ n (U) so liminfμ n (U) > 0 which implies that x liminf(supp(μ) n ) which implies that Supp(ν) liminfsupp(μ n ) therefore liminf(supp(μ) n ).

11 Weak convergence of convolution products 119 We now solve the Choquet equation for not necessarily commutative hypergroups (an alternative proof can also be found in [BH95] but required lots of steps). Corallory 3.1 Suppose H is a hypergroup with an invariant measure and μ, ν M 1 (H). Then μ = μ ν if and only if μ = μ δ x for all x [Supp(ν)](the smallest subhypergroup of H containing Supp(ν) ) Proof: The if part is trivial. Now suppose that μ = μ ν then μ = μ ν n. Given ɛ>0, let K be a compact subset of H such that μ(k) > 1 ɛ Then 1 ɛ<μ(k) =μ = μ ν n (K) = δ x ν n (K)μ(dx) ν n (x K)μ(dx) ν n (x K)μ(dx)+ɛ ν n (K K)+ɛ Where K K = x K x K. Since K K H, K K is compact, and consequently the sequence ν n is tight. We can now use theorem ( 3.1)since 1 n n k=1 νk converges weakly to some idempotent probability measure β. Also since μ = μ ν n we have μ = μ ( 1 n n k=1 νk )and since convolution is separately continuous with respect to weak topology we have μ = μ β, where β = β β, and consequently, Supp(β) is a compact subhypergroup of H [[Je75] theorem 10.2E] containing Supp(ν). And since ν β = β ν = β (β is the Haar measure of [Supp(β)]). Now suppose μ = μ β Let f C c (H) and define g by g(x) =μ δ x (f) for all x H then β δ x (g) = g(y)β δ x (dy) = μ δ y (f)β δ x (dy) = f(z y)μ(dz)β δ x (dy) = μ β δ x (f) =μ δ x (f) =g(x) Since g C 0 (H) and Supp(β)is compact, there exists x 0 Supp(β) such that g(x 0 )= g Supp(β) = Sup x Supp(β) g(x) Now β δ x0 (g) =g(x 0 ) so that g(x 0 )= g(y x 0 )β(dy) which implies g(x 0 )= g(y x 0 ) for all y Supp(β) which implies g(x 0 )=g(y x 0 )= g(u)δ y δ x0 (du) =

12 120 Norbert Youmbi μ δ u (f)δ y δ x0 (du) = f(z u)μ(dz)δ y δ x0 (du) = μ δ y δ x0 (f) Since g(x 0 )=g(y x 0 ) for all y Supp(β) g is constant on Supp(β) x 0 Supp(β) which is a right ideal of Supp(β) so contains the neutral element e. So we have g(e) =μ δ y δ e (f) =μ δ y (f) and since g(e) =μ δ e (f) we have μ δ e (f) =μ δ y (f) so that μ(f) =μ δ y (f) for all f C c (H). Therefore μ = μ δ y for all y Supp(β) and since Supp(ν) Supp(β) we have that μ = μ δ x for all x [Supp(ν)] Corallory 3.2 Let S be a semihypergroup and ν M 1 (S) be such that the sequence (ν n ) is tight and S = [ n=1 Supp(ν)n ]. Let μ M 1 (S) such that μ ν = μ Then the following assertions are valid. i. S has a simple ideal K = Supp(ν 0 ), where ν 0 is the weak limit of 1 n n k=1 νk and ν ν 0 = ν 0 ν = ν 0 ii. Supp(μ) K and μ = μ μ Proof: Assertion (i) follows from theorem ( 3.1). Suppose now that μ ν = μ for some μ M 1 (S). Then μ ( 1 n n ν k )=μ, n 1 k=1 and it follows that μ ν 0 = μ and Supp(μ) =Supp(μ)Supp(ν 0 ) Supp(ν 0 )= K Now let x Supp(μ) and f C b (H) then μ ν 0 = μ implies δ x μ(f) =δ x μ ν 0 (f) = δ x δ y ν 0 (f)μ(dy) = δ x ν 0 (f)μ(dy) =δ x ν 0 (f)

13 Weak convergence of convolution products 121 We have δ x δ y ν 0 = δ x ν 0 since Supp(μ) Supp(ν 0 ) and ν 0 = ν 0 ν 0 by proposition (??).And it follows that μ μ(f) = δ x μ(f)μ(dx) = δ x ν 0 (f)μ(dx) =μ ν 0 (f) =μ(f) So that μ = μ μ. is an idempotent measure so Supp(μ) is a simple subsemihypergroup of K = Supp(ν 0 ) Corallory 3.3 Let S be a semihypergroup and ν M 1 (S),S = [ n=1 Supp(ν)n ]. Suppose that S satisfies the following compactness condition K is compact, x S = x K is compact Let μ M 1 (S) such that μ ν = μ then 1 n n k=1 νk converges weakly to ν 0 M 1, and consequently all the results in corollary 3.3 remain valid. Proof: Let λ be a weak* limit points of the sequence ν n = 1 n n k=1 ν k If all such weak* limit points are probability measures, then it follows from theorem 3.1, that the sequence 1 n n k=1 νk converges weakly to some ν 0 in M 1 (S), and the rest of corollary 3.3 then follows exactly as in corollary 3.2. Thus it suffices to show that λ M 1 (S). Let f C c (S) and x S. Then f x C c (S). Let (n k ) be the subsequence such that (ν nk ) weak* converges to λ. Then let us define the function g k and g by g k (x) =δ x ν nk (f) and g(x) =δ x λ(f) Since convolution is separately continuous δ x ν nk w δ x λ, sog k (x) g(x) as k therefore by the bounded convergence theorem, for f C c (S) we have μ(f) = f(x)μ(dx) =μ ν nk (f) = δ x ν nk (f)μ(dx) = g k (x)μ(dx) g(x)μ(dx) =

14 122 Norbert Youmbi δ x λ(f)μ(dx) =(μ λ)(f) So that μ = μ λ. That is μ(s) =μ(s)λ(s) which implies that λ(s) =1so λ M 1 (S). Theorem 3.2 Suppose S is a compact semihypergroup, with a continuous concretization, μ M 1 (S) and S = [ n=1 Supp(μ)n ] then for any open set G containing the kernel K of S, lim n μ n (G) =1 Proof: Let K G, G open, since K S G, S, K are compact, there exists an open set V containing K such that V S G. Notice that if lim k μ n k (V )=1, (2) then ɛ >0 there exists k 0 such that m>n k0 implies μ m (G) μ n k 0(V )μ m n k0 (S) > 1 ɛ which means that lim n μ n (G) =1 Therefore it is enough to established ( 2) for some subsequence (n k ). To this end let x K then since SxS K V there exists an open set W such that x W and S W S V since x W S = [ n=1 Supp(μ)n ] there exists m>0 such that μ m (W ) > 0. Let (X n ) be a sequence of independent S-valued random variable each with distribution μ m. Then we have P (Xn W )= and by Borel Cantelli lemma we have P (X n W, i.o) =1 Since {X n W } are independent, ɛ >0 m 0 such that m 0 P ( {X n W } > 1 ɛ. n=0 Now if m 0 x {X n W }. n=0

15 Weak convergence of convolution products 123 n 0 such that X n0 (x) W, let S n = n k=1 ˆX k n m 0. Note that if X and Y are two random variables such that X is A-valued and Y is B-valued then X ˆ+Y is AB-valued. For X ˆ+Y =Φ(X, Y, ξ) when ξ is [0, 1]-valued so that (X ˆ+Y )(x) = Φ(X(x),Y(x),ξ(t)). Set X(x) =z, Y (x) =y, ξ(t) =s Claim: Φ(z,y, s) Supp(δ z δ y ) A B for all s [0, 1].To see this suppose x A, y B let V be an open set containing Φ(z,y, s), s [0, 1] then δ x δ y (V )=λ{s :Φ(z,y, s) V } and since Φ is continuous, λ{s :Φ(z,y, s) V } > 0 so that δ x δ y (V ) > 0, that is Φ(z,y, s) Supp(δ z δ y ) A B. So X ˆ+Y is AB-valued and by the definition of the randomized sum Since X n0 S n = X 1 ˆ+X 2 ˆ+X 3 ˆ+... ˆ+X n is K-valued S n will be V -valued so that m 0 {X n W } {X 1 ˆ+X 2 ˆ+X 3 ˆ+... ˆ+X n V },n m 0 n=0 Since S W S V. Now as X 1 ˆ+X 2 ˆ+X 3 ˆ+... ˆ+X n has distribution μ mn, it is clear that for n m 0 (mn > m 0 ) so that μ mn (V ) > 1 ɛ for all ɛ>0. So μ mn (V )=1. Proposition 3.1 Let I be a Borel set that is an ideal of S. Suppose that for some positive integer m, μ m (I) > 0 for some μ M 1 (S). Then the sequence (μ n (I)) monotonically increases to 1. Proof: I S I so μ n+1 (I) μ n (I)μ(S) =μ n (I) For all positive integer n. Now the prove follows as above since S I S I. Theorem 3.3 Let μ n be a sequence in M 1 (S) such that the subsequence μ 0,nt where μ k,n = μ k+1... μ n has at least one weak* limit point in M 1 (S). Suppose that S has the property such that convolution as a map from M 1 (S) B(S) B(S) is continuous in the weak* sense. Then there is a sequence (p t ) (n t ) such that for each positive integer k μ k,pt w λ k M 1 (S) λ pt w λ λ M 1 (S) λ k λ = λ k Where B(S) ={μ : μ is a nonnegative regular Borel measure with μ(s) 1}

16 124 Norbert Youmbi Proof: Suppose μ 0,nt w λ 0 M 1 (S). Note that w*-convergence is weak convergence when the limit is in M 1 (S). Now for each positive integer t y nt (μ 0,nt,μ 1,nt,...,μ nt 1,n t, 0, 0, 0,...) are elements in the product space Y = X i, i=1 X i = B(S) with weak*topology, where Y has the product topology and is therefore compact, since B(S) is w*compact. Since Y is compact (and first countable), there is a subsequence (m t ) (n t ) such that y mt y Y, in the topology of Y. This means that for each k 0, there exists λ k B(S) such that μ k,mt λ k Since convolution is continuous as a map from M 1 (S) B(S) B(S) it follows that for each k 1 μ 0,mt = μ 1 μ 2...mu k μ k+1... μ mt = μ 0,k μ k,mt μ 0,k λ k in the weak* sense and this means that μ 0,k λ k = λ 0,k 1 (since μ 0,mt λ 0 ) However since λ 0 M 1 (S) this implies that λ k M 1 (S) for each k 1. Let (p t ) (m t ) be a subsequence such that λ pt λ B(S) in the weak*sens. Now for fixed integer s and t>ssuch that p s >k, we have μ k,ps μ ps,p t = μ k,pt Again by the continuity of convolution, it follows that given k 0 for each s such that p s >k μ k,ps λ ps = λ k which in turn implies that λ k λ = λ k, k 1, since λ k M 1 (S). The last equation implies that λ λ = λ. Proposition 3.2 Suppose S satisfies the following compactness condition. K compact and x S implies x K is compact. If μ n μ weakly in M 1 (S) and ν n ν B(S) in the weak* sense with ν n M 1 (S) then μ n ν n μ ν in the weak* topology.

17 Weak convergence of convolution products 125 Proof: Let f C c (S). Then for each s S, t f s (t) isinc c (S). Hence if g n (s) f(s t)ν n (dt) g(s) f(s t)ν(dt) Then lim n g n (s) =g(s) Since ν is a regular measure, it is easily seen that g is a bounded continuous function on S. Also by Egoroff s theorem in analysis, given ɛ>0 there exists a compact set K such that μ(k) <ɛand on S K, g n g uniformly. Since μ n μ weakly, limsup n μ n (K) μ(k) <ɛ Then we have g n dμ n gdμ g n dμ n gdμ n + g n g dμ n + gdμ n gdμ K K K c which shows that lim n g n dμ n = gdμ because g n dμ n gdμ = g n dμ n gdμ n + gdμ n gdμ = g n dμ n + g n dμ n gdμ n gdμ n + gμ n gdμ. = K K c K K c g n dμ n gdμ n + (g n g)dμ n + gdμ n gdμ K K K c So lim n g n dμ n = gdμ This means that fdμ n ν n = f(s t)μ n (ds)ν n (dt) = [ f(s t)ν n (dt)]μ n (ds) = g n (s)μ n (ds) gdμ = fdμ ν So μ n ν n μ ν

18 126 Norbert Youmbi References [BH95] [Du73] [HR70] [HK75] [HM95] [Je75] [Sp78] [Yo12] Bloom W.R and Heyer H.,The Harmonic Analysis of Probability Measures on Hypergroups, de Gruyter Stud. Math.,vol.20, de Gruyter, Berlin and Hawthorne, New York, Dunkl C.F.,The measure Algebra of a Locally Compact Hypergroup, Trans. Amer. Math. Soc. 179(1973), Hewitt E. and Ross A. K., Abstract Harmonic Analysis, Vol2 Springer-Verlag, Berlin and New York, 1970.MR Hewitt E. and Karl S., Real and Abstract Analysis, Third printing Graduate Text in math. 25 Springer Verlag, New York- Heidelberg (1975) Högnäs G. and Mukherjea A. Probability Measures on Semigroups, Convolution Products, Random Walks, and Random matricesplenum Press, New York and London. Jewett R.I.,Spaces with an abstract convolution of measures, Advances in Math. 18(1975), Spector R.,Mésures invariantes sur les hypergroupes, Trans. Amer. Soc. 239(1978) Youmbi N., A Rees convolution product for topological semihypergroups, Int. Math. Forum Vol. 7, No (2012) [Ze89] Zeuner H. One dimensional hypergroups Adv. math., 76:1-18. Received: December 27, 2013