Int. Journal of Math. Analysis, Vol. 8, 2014, no. 3, 109 - 126
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ijma.2014.312304
Weak Convergence of Convolution Products of
Probability Measures on Semihypergroups
Norbert Youmbi
Department Of Mathematics
Saint Francis University
Loretto, PA 15940-0600 USA
c 2014 Norbert Youmbi. This is an open access article distributed under the
Copyright 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
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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; Cb (S): the space of bounded elements of C(S); C0 (S): the
space of elements of Cb (S) which tends to 0 at ∞; Cc (S): the space of elements
of C0 (S) with compact support; Cc+ (S): the space of nonnegative elements of
Cc (S); M(S) denotes the set of finite regular Borel measures; M+ (S) the set
of non-negative measures; M1 (S) denote the set of probability measures; If
μ ∈ M(S) then Supp(μ) = {x ∈ S : if V 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:
(SH1 ) (M(S), +, ∗) is a Banach algebra.
(SH2 ) For all x, y ∈ S, δx ∗ δy is a probability measure with compact support.
(SH3 ) The mapping (x, y) → δx ∗ δy of S × S into M1 (S), where S × S has
the product topology and M1 (S) has the weak topology, is continuous.
(SH4 ) The mapping(x, y) → Supp(δx ∗ δy ) 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 CU (V ) =
{C ∈ C(S) : C ∩ U = ∅ and C ⊂ V } where U and V are open subsets of
S.
If in addition,
SH5 there exists e ∈ S such that δx ∗ δe = δe ∗ δx = δx ∀x ∈ S, and
SH6 There exists a topological involution (a homeomorphism) from S onto
S such that (x− )− = x ∀x ∈ S, with (δx ∗ δy )− = δy− ∗ δx− and e ∈
Supp(δx ∗ δy ) 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 δx ∗ δy = δy ∗ δx for all x, y ∈ S we say that (S, ∗) is a
commutative semihypergroup.
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Weak convergence of convolution products
(ii) The convolution ∗ on M(S) is defined by
f dμ ∗ ν =
μ(dx) ν(dy) f dδx ∗ δy .
μ ∗ ν(f ) =
S
S
S
S
for all f ∈ Cb (S).
Example 1.1
define
1. Let S = {e, a, b}. Let e be the identity element and let us
1
1
δa ∗ δa = δa + δb
2
2
δb ∗ δb = δa
1
1
δa ∗ δb = δb ∗ δa = δe + δb
2
2
Then (S, ∗) is a semihypergroup and if we defined an involution by a− = b
and b− = a we have
1
1
1
1
(δa ∗ δa )− = δa− + δb− = δb + δa
2
2
2
2
But
1
1
δa− ∗ δa− = δb ∗ δb = δa = δa + δb ,
2
2
although e ∈ Supp(δa ∗ δb ) this involution does not satisfy the condition (δa ∗ δb )− = δb− ∗ δa− 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
δx ∗ δx = aδe + bδx + cδy
δy ∗ δy = a δe + c δx + b δy
δx ∗ δy = δy ∗ δx = qδx + q δy
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 1.2
1. An element e ∈ S is called a left (right) identity element
of S if δe ∗ δx = δx ( δx ∗ δe = δx ) 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 δz ∗ δx = δz
(δx ∗ δz = δz ) for all x ∈ S. If z is both left and right zero, we simply call
it the zero of S. A semihypergroup has at most one zero.
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Norbert Youmbi
3. An element a ∈ S is called an idempotent element of S if δa ∗ δa = δa
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(δx ∗ δy ) 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 : δx ∗ δx− = δx− ∗ δx = δe }.
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
δx ∗ δx = δy
1
3
δy ∗ δy = δx + δy
4
4
1
1
δx ∗ δy = δy ∗ δx = δx + δy
2
2
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
1
1
δr ∗ δs = δ|r−s| + δ1−|1−r−s|
2
2
for all r, s ∈ [0, 1] Zeuner [Ze89] proved that ([0, 1], ∗) is a hypergroup
with a nontrivial center {0, 1}.
Definition 1.4
1. 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}).
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Weak convergence of convolution products
3. ∀a, b ∈ S we say that the equation xa = b is solvable if and only if there
exists x0 ∈ S such that b ∈ Supp(δx0 ∗ δa )
Definition 1.5
1. 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 ⇐⇒ δe ∗ δf = δf ∗ δe = δe
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, δa is not necessarily a primitive idempotent in M1 (S), according
to the definition of primitive idempotents in the semigroup (with respect to
convolution)M1 (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 Cc (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 (μα ) in B(S), w∗ converges to μ in B(S) if and only if for every f in Cc (S), f dμα → f dμ.
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 > 0, there is a compact subset K ⊂ S such that
μ ∈ Γ ⇒ μ(K ) > 1 − 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) is w ∗ -compact.
Definition 1.7 If f is a Borel function on S and x, y ∈ S, then we define
y
f d(δx ∗ δy )
f (x ∗ y) ≡ fx (y) ≡ f (x) =
S
If this integral exists, even when it is not finite, fx is called the left translation
of f and f x is called the right translation of f .
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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. fx 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. fx and f x are Borel functions in S
iii.
iv.
f d(μ ∗ ν) =
S
f dμ =
S x
S
S
S
f (x ∗ y)μ(dx)ν(dy)
f d(δx ∗ μ)
v. fx (y ∗ z) = f z (x ∗ y)
Notation 1.1 Let S be a locally compact semihypergroup.
Then ∀x ∈ S, μ ∈ M1 (S), and f ∈ C(S), we write:
δx ∗ μ(f ) =
S
fx dμ
(≡ μ(fx ), say)
and also,
μ ∗ δx (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(δy ∗ δx ) ∩ B = ∅}
Similarly,
x− B = {y ∈ S : Supp(δx ∗ δy ) ∩ 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.
Weak convergence of convolution products
2
115
Concretization for Semihypergroups
Definition 2.1 A triplet (X, μ, Φ) consisting of a compact space X, a probability measure μ ∈ M1 (X), and a Borel-measurable mapping Φ : S ×S ×X → S
is called a concretization of the semihypergroup (S, ∗) if
μ({z ∈ X : φ(x, y, z) ∈ A}) = δx ∗ δy (A)
For all x, y ∈ S and A ∈ B(S).
Example 2.1
1. 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 = δe 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
1
1
δx ∗ δy = δ|x−y| + δx+y
2
2
for all x, y ∈ K we obtain the concretization (X, μ, Φ) where
1
1
X = {−1, 1}, μ = δ−1 + δ1
2
2
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}) = δx ∗ δy (A)
Actually
μ({z ∈ X : φ(x, y, z) ∈ A}) =
1
1
δx ∗ δy (A) = δ|x−y| (A) + δx+y (A)
2
2
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
/ A, then
and δx ∗ δy (A) = 0. If Φ(x, y, −1) ∈ A and Φ(x, y, 1) ∈
/ A
μ({−1}) = 12 and δx ∗ δy (A) = 12 δ|x−y| (A) = 12 and if Φ(x, y, −1) ∈
1
1
and Φ(x, y, 1) ∈ A then μ({1}) = 2 and δx ∗ δy (A) = 2 δx+y (A) = 12 .
Finally if Φ(x, y, −1) ∈ A and Φ(x, y, 1) ∈ A then μ({−1, 1}) = 1 and
δx ∗ δy (A) = 1. So we have (X, μ, Φ) as defined above is a concretization
of (R+ , ∗).
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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(δx ∗ δy ) = |x − y|
maxsupp(δx ∗ δy ) = 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 (Xn )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 Pξ = μ we define the randomized sum of X with Y by X +̂Y = Φ(X, Y, ξ).
Remark 2.2 This definition can be extended to sequences (Xn )n≥1 of Xvalued random variables on (Ω, A, P ) provided all random variables occurring
in the sequence (Xn )n≥1 and (ξn )n≥1 are independent and Pξn := μ for all n ≥ 1
in fact by the recurrence
0
X̂j := e
j=1
n
j=1
X̂j := Xn +̂
n−1
j=1
X̂j , n ≥ 1
Weak convergence of convolution products
117
the randomized sums Sn = nj=1 X̂j , n ≥ 1 are introduced again as S-valued
random variables on (Ω, A, P ), which form a (non homogeneous) Markov chain
(Sn )n≥0 with corresponding sequence (Nn )n≥1 of transition kernels on (S, B(S))
satisfying
Nn (x, A) = (PXn ∗ δx )(A) =
P (Sn ∈ A : Sn−1 = x)
For PSn−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 Pξn := μ such that X, Y, ξ are
independent then PX +̂Y = PX ∗ PY
Proof:
∀A ∈ (S, B(S))
PX +̂Y (A) = P (Φ(X, Y, ξ) ∈ A) =
P (Φ(X, Y, ξ) ∈ A)PX (dx)PY (dy)
μ[Φ(X, Y, ξ) ∈ A)]PX (dx)PY (dy)
δx ∗ δy (A)PX (dx)PY (dy)
PX ∗ PY (A)
So PX +̂Y = PX ∗ PY
Remark 2.3 Forming randomized sums is generally not an associative operation although convolution obviously is. While randomized sum X +̂Y clearly
depends on the particular choice of the underlying concretization of S the joint
distribution of the random variables X, Y and X +̂Y does not.
3
Sequence of Convolution Powers of Probability Measures
Theorem 3.1 Let S be a locally compact semihypergroup. Assume μ ∈ M1 (S)
and suppose that the sequence (μn ) is tight. Suppose also that
S=[
∞
n=1
Supp(μ)n]
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Norbert Youmbi
let K = {μ ∈ M1 (S) : μ is a weak limit point of the sequence (μ)n } also let us
define
S0 = {Supp(λ) : λ ∈ K}
and S1 = S¯0 then the sequence
1 k
μ
n k=1
n
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 μn =
1
n
n
k=1 μ
k
then for k ≥ 1, μk ∗ μn = μn ∗ μk and
limn→∞ μn − μk ∗ μn = 0
(1)
its follows , since the sequence μn is tight, that the sequence (μn ) is also tight
so that {(μn ) : n ≥ 1} is weakly relatively compact. Let ν1 and ν2 be two limit
points of (μn ) then by ( 1)
μk ∗ ν1 = ν1 ∗ μk = ν1
μk ∗ ν2 = ν2 ∗ μk = ν2
It follows that
and
1
1
μk ∗ ν1 =
ν1 ∗ μk = ν1
n k=1
n k=1
n
n
1
1
μk ∗ ν2 =
ν2 ∗ μk = ν2
n k=1
n k=1
n
n
That is
μn ∗ ν1 = ν1 ∗ μn = ν1
μn ∗ ν2 = ν2 ∗ μn = ν2
which then implies that ν1 = ν2 (≡ 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
n
ideal of S = [ ∞
n=1 Supp(μ) ]
Remark 3.1 If μn converges weakly then liminf(Supp(μ)n ) is nonempty. To
see this suppose μn −→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) ≤
liminfμn (U) so liminfμn (U) > 0 which implies that x ∈ liminf(Supp(μ)n )
which implies that Supp(ν) ⊂ liminfSupp(μn ) therefore liminf(Supp(μ)n ) =
∅.
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Weak convergence of convolution products
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 ∈ M1 (H). Then μ = μ∗ν if and only if μ = μ∗δx 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
> 0, let K be a compact subset of H such that μ(K) > 1 − Then
n
1 − < μ(K) = μ = μ ∗ ν (K) = δx ∗ ν n (K)μ(dx) ≤
n
−
ν (x K)μ(dx) ≤
ν n (x− K)μ(dx) + ≤
ν n (K − ∗ K) + 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
n
1
k
probability measure β. Also
k=1 ν converges weakly to some
n
nidempotent
1
n
k
since μ = μ∗ν we have μ = μ∗( n k=1 ν )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 ∈ Cc (H) and define g by
g(x) = μ ∗ δx (f ) for all x ∈ H then
β ∗ δx (g) = g(y)β ∗ δx (dy) = μ ∗ δy (f )β ∗ δx (dy) =
f (z ∗ y)μ(dz)β ∗ δx (dy) =
μ ∗ β ∗ δx (f ) = μ ∗ δx (f ) = g(x)
Since g ∈ C0 (H) and Supp(β)is compact, there exists x0 ∈ Supp(β) such that
g(x0 ) = gSupp(β) = Supx∈Supp(β) |g(x)|
Now β ∗ δx0 (g) = g(x0 ) so that g(x0 ) = g(y ∗ x0 )β(dy) which implies g(x0 ) =
g(y ∗ x0 ) for all y ∈ Supp(β) which implies
g(x0 ) = g(y ∗ x0 ) = g(u)δy ∗ δx0 (du) =
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Norbert Youmbi
μ ∗ δu (f )δy ∗ δx0 (du) =
f (z ∗ u)μ(dz)δy ∗ δx0 (du) =
μ ∗ δy ∗ δx0 (f )
Since g(x0 ) = g(y ∗ x0 ) for all y ∈ Supp(β) g is constant on Supp(β) ∗ x0 ⊂
Supp(β) which is a right ideal of Supp(β) so contains the neutral element e.
So we have
g(e) = μ ∗ δy ∗ δe (f ) = μ ∗ δy (f )
and since g(e) = μ ∗ δe (f ) we have
μ ∗ δe (f ) = μ ∗ δy ∗ (f )
so that μ(f ) = μ ∗ δy (f ) for all f ∈ Cc (H). Therefore μ = μ ∗ δy for all
y ∈ Supp(β) and since Supp(ν) ⊂ Supp(β) we have that μ = μ ∗ δx for all
x ∈ [Supp(ν)]
Corallory 3.2 Let S be a semihypergroup and ν ∈ M1 (S) be such that the
n
sequence (ν n ) is tight and S = [ ∞
n=1 Supp(ν) ]. Let μ ∈ M1 (S) such that
μ ∗ ν = μ Then the following assertions are valid.
i. S has a simple ideal K = Supp(ν0 ), where ν0 is the weak limit of n1 nk=1 ν k
and ν ∗ ν0 = ν0 ∗ ν = ν0
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 ∈ M1 (S). Then
1 k
ν ) = μ, n ≥ 1
μ∗(
n k=1
n
and it follows that μ ∗ ν0 = μ and Supp(μ) = Supp(μ)Supp(ν0) ⊂ Supp(ν0 ) =
K
Now let x ∈ Supp(μ) and f ∈ Cb (H) then μ ∗ ν0 = μ implies
δx ∗ μ(f ) = δx ∗ μ ∗ ν0 (f ) =
δx ∗ δy ∗ ν0 (f )μ(dy) =
δx ∗ ν0 (f )μ(dy) = δx ∗ ν0 (f )
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Weak convergence of convolution products
We have δx ∗ δy ∗ ν0 = δx ∗ ν0 since Supp(μ) ⊂ Supp(ν0 ) and ν0 = ν0 ∗ ν0 by
proposition ( ??).And it follows that
μ ∗ μ(f ) = δx ∗ μ(f )μ(dx) =
δx ∗ ν0 (f )μ(dx) = μ ∗ ν0 (f ) = μ(f )
So that μ = μ ∗ μ. is an idempotent measure so Supp(μ) is a simple subsemihypergroup of K = Supp(ν0 )
n
Corallory 3.3 Let S be a semihypergroup and ν ∈ M1 (S),S = [ ∞
n=1 Supp(ν) ].
Suppose that S satisfies the following compactness condition
K is compact, x ∈ S =⇒ x− K is compact Let μ ∈ M1 (S) such that μ ∗ ν = μ then n1 nk=1 ν k converges weakly to
ν0 ∈ M1 , and consequently all the results in corollary 3.3 remain valid.
Proof:
Let λ be a weak* limit points of the sequence
1 k
ν
νn =
n k=1
n
If all such weak* limit points are probability
measures, then it follows from
n
1
k
theorem 3.1, that the sequence n k=1 ν converges weakly to some ν0 in
M1 (S), and the rest of corollary 3.3 then follows exactly as in corollary 3.2.
Thus it suffices to show that λ ∈ M1 (S).
Let f ∈ Cc (S) and x ∈ S. Then fx ∈ Cc (S). Let (nk ) be the subsequence
such that (νnk ) weak* converges to λ. Then let us define the function gk and
g by
gk (x) = δx ∗ νnk (f )
and
g(x) = δx ∗ λ(f )
Since convolution is separately continuous δx ∗νnk →w∗ δx ∗λ, so gk (x) −→ g(x)
as k −→ ∞ therefore by the bounded convergence theorem, for f ∈ Cc (S) we
have
μ(f ) = f (x)μ(dx) = μ ∗ νnk (f ) =
δx ∗ νnk (f )μ(dx) =
gk (x)μ(dx) −→
g(x)μ(dx) =
122
Norbert Youmbi
δx ∗ 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) = 1 so
λ ∈ M1 (S).
Theorem 3.2 Suppose S is a compact semihypergroup, with a continuous concretization, μ ∈ M1 (S) and
n
S=[ ∞
n=1 Supp(μ) ] then for any open set G containing the kernel K of S,
limn→∞ μn (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
limk→∞ μnk (V ) = 1,
(2)
then ∀ > 0 there exists k0 such that m > nk0 implies
μm (G) ≥ μnk0 (V )μm−nk0 (S) > 1 − which means that
limn→∞ μn (G) = 1
Therefore it is enough to established ( 2) for some subsequence (nk ). To this
end let x ∈ K then since SxS ⊂ K ⊂ V there exists
an open setn W such that
x ∈ W and S ∗ W ∗ S ⊂ V since x ∈ W ⊂ S = [ ∞
n=1 Supp(μ) ] there exists
m
m > 0 such that μ (W ) > 0.
Let (Xn ) be a sequence of independent S-valued random variable each with
distribution μm . Then we have
P (Xn ∈ W ) = ∞
and by Borel Cantelli lemma we have
P (Xn ∈ W, i.o) = 1
Since {Xn ∈ W } are independent, ∀ > 0 ∃m0 such that
P(
m0
{Xn ∈ W } > 1 − .
n=0
Now if
x∈
m0
n=0
{Xn ∈ W }.
Weak convergence of convolution products
123
∃n0 such that Xn0 (x) ∈ W , let Sn = nk=1 X̂k n ≥ m0 .
Note that if X and Y are two random variables such that X is A-valued
and Y is B-valued then X +̂Y is AB-valued. For X +̂Y = Φ(X, Y, ξ) when ξ
is [0, 1]-valued so that (X +̂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(δz ∗ δy ) ⊂ 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
δx ∗ δy (V ) = λ{s : Φ(z, y, s) ∈ V } and since Φ is continuous, λ{s : Φ(z, y, s) ∈
V } > 0 so that δx ∗ δy (V ) > 0, that is Φ(z, y, s) ∈ Supp(δz ∗ δy ) ⊂ A ∗ B. So
X +̂Y is AB-valued and by the definition of the randomized sum
Sn = X1 +̂X2 +̂X3 +̂ . . . +̂Xn
Since Xn0 is K-valued Sn will be V -valued so that
m0
{Xn ∈ W } ⊂ {X1 +̂X2 +̂X3 +̂ . . . +̂Xn ∈ V }, n ≥ m0
n=0
Since S ∗ W ∗ S ⊂ V .
Now as X1 +̂X2 +̂X3 +̂ . . . +̂Xn has distribution μmn , it is clear that for
n ≥ m0 (mn > m0 ) so that μmn (V ) > 1 − for all > 0. So μmn (V ) = 1.
Proposition 3.1 Let I be a Borel set that is an ideal of S. Suppose that for
some positive integer m, μm (I) > 0 for some μ ∈ M1 (S). Then the sequence
(μn (I)) monotonically increases to 1.
Proof:
I ∗ S ⊂ I so μn+1 (I) ≥ μn (I)μ(S) = μn (I)
For all positive integer n. Now the prove follows as above since S ∗I ∗S ⊂ I.
Theorem 3.3 Let μn be a sequence in M1 (S) such that the subsequence μ0,nt
where
μk,n = μk+1 ∗ . . . ∗ μn
has at least one weak* limit point in M1 (S). Suppose that S has the property
such that convolution as a map from M1 (S) × B(S) −→ B(S) is continuous
in the weak* sense. Then there is a sequence (pt ) ⊂ (nt ) such that for each
positive integer k
μk,pt −→w λk ∈ M1 (S)
λpt −→w λ ∗ λ ∈ M1 (S)
λk ∗ λ = λk
Where B(S) = {μ : μ is a nonnegative regular Borel measure with μ(S) ≤ 1}
124
Norbert Youmbi
Proof:
Suppose μ0,nt −→w λ0 ∈ M1 (S). Note that w*-convergence is weak convergence when the limit is in M1 (S). Now for each positive integer t
ynt ≡ (μ0,nt , μ1,nt , . . . , μnt−1 ,nt , 0, 0, 0, . . .)
are elements in the product space
Y =
∞
Xi ,
Xi = B(S)
i=1
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 (mt ) ⊂ (nt ) such that ymt −→ y ∈ Y , in the topology of Y .
This means that for each k ≥ 0, there exists λk ∈ B(S) such that
μk,mt −→ λk
Since convolution is continuous as a map from M1 (S) × B(S) −→ B(S) it
follows that for each k ≥ 1
μ0,mt = μ1 ∗ μ2 ∗ . . . muk ∗ μk+1 ∗ . . . ∗ μmt =
μ0,k ∗ μk,mt −→ μ0,k ∗ λk
in the weak* sense and this means that
μ0,k ∗ λk = λ0 , k ≥ 1
(since μ0,mt −→ λ0 )
However since λ0 ∈ M1 (S) this implies that λk ∈ M1 (S) for each k ≥ 1. Let
(pt ) ⊂ (mt ) be a subsequence such that λpt −→ λ ∈ B(S) in the weak*sens.
Now for fixed integer s and t > s such that ps > k, we have
μk,ps ∗ μps,pt = μk,pt
Again by the continuity of convolution, it follows that given k ≥ 0 for each s
such that ps > k
μk,ps ∗ λps = λk
which in turn implies that λk ∗ λ = λk , k ≥ 1, since λk ∈ M1 (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 μn −→ μ weakly in M1 (S) and νn −→ ν ∈ B(S) in the weak* sense with
νn ∈ M1 (S) then μn ∗ νn −→ μ ∗ ν in the weak* topology.
125
Weak convergence of convolution products
Proof:
Let f ∈ Cc (S). Then for each s ∈ S, t → fs (t) is in Cc (S). Hence if
gn (s) ≡ f (s ∗ t)νn (dt)
g(s) ≡
f (s ∗ t)ν(dt)
Then
limn→∞ gn (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 > 0 there exists
a compact set K such that μ(K) < and on S − K, gn −→ g uniformly. Since
μn −→ μ weakly,
limsupn→∞ μn (K) ≤ μ(K) < gn dμn − gdμ ≤
gn dμn −
gdμn +
|gn − g| dμn + gdμn − gdμ
c
Then we have
K
K
K
because
gn dμn −
gn dμn =
gdμ =
K
limn→∞
which shows that
gn dμn −
gdμ
gdμn +
gdμn −
gdμ =
gn dμn +
gn dμn −
gdμn −
gdμn + gμn − gdμ. =
Kc
K
Kc
gn dμn −
gdμn +
(gn − g)dμn + gdμn − gdμ
K
Kc
K
limn→∞
This means that
So
gn dμn =
gdμ
f dμn ∗ νn =
f (s ∗ t)μn (ds)νn (dt) =
[ f (s ∗ t)νn (dt)]μn (ds) =
gn (s)μn (ds) −→ gdμ = f dμ ∗ ν
So μn ∗ νn −→ μ ∗ ν
126
Norbert Youmbi
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Received: December 27, 2013