Equipment of sets with cardinality of the continuum by structures of Polish groups with Haar measures

It is introduced a certain approach for equipment of sets with cardinality of the continuum by structures of Polish groups with two-sided (left or right) invariant Haar measures. By using this approach we answer positively Maleki’s certain question(2012) what are the real k-dimensional manifolds with at least two different Lie group structures that have the same Haar measure. It is demonstrated that for each diffused Borel probability measure μ defined in a Polish space (G, ρ,Bρ(G)) without isolated points there exist a metric ρ1 and a group operation⊙ inG such that Bρ(G) = Bρ1(G) and (G, ρ1,Bρ1(G),⊙) stands a compact Polish group with a two-sided (left or right) invariant Haar measure μ, where Bρ(G) and Bρ1(G) denote Borel σ-algebras of subsets of G generated by metrics ρ and ρ1, respectively. Similar approach is used for a construction of locally compact non-compact or non-locally compact Polish groups equipped with two-sided (left or right) invariant quasi-finite Borel measures.

implies that the answer to Problem 1.1 is yes. The measure µ satisfying conditions (i)-(v) is called a left (right or two-sided) invariant Haar measure in a locally compact Polish group (G, ρ, ⊙).
In this note we would like to study the following problems, which can be considered as converse (in some sense) to Problem 1.1. Problem 1.2. Let (G, ρ) be a Polish metric space which is dense-in-itself. Let µ be a diffused Borel probability measure defined in (G, ρ). Do there exist a metric ρ 1 and a group operation ⊙ in G such that the following three conditions (j) The class of Borel measurable subsets of G generated by the metric ρ 1 coincides with the class of Borel measurable subsets of the same space generated by the metric ρ, (jj) (G, ρ 1 , ⊙) is a compact Polish group and (jjj) µ is a left(right or two-sided) invariant Haar measure in (G, ρ 1 , ⊙) hold true ?
For a solution of the Problem 1.2 , there will be used especially an existence of Borel isomorphism between two diffused Borel probability measures in Polish groups (cf. [3]). Problem 1.3. Let (G, ρ) be a Polish metric space which is dense-in-itself. Let µ be a diffused σ-finite non-finite Borel measure defined in (G, ρ). Do there exist a metric ρ φ , a group operation ⊙ φ in G and the Borel measure µ ⋆ in G such that the following four conditions (i) The class of Borel measurable subsets of G generated by the metric ρ φ coincides with the class of Borel measurable subsets of the same space generated by the metric ρ, (ii) (G, ρ φ , ⊙ φ ) is a non-compact locally compact Polish group, (iii) The measures µ ⋆ and µ are equivalent and (iv) µ ⋆ is a left (right or two-sided) invariant σ-finite non-finite Haar measure in (G, ρ φ , ⊙ φ ) hold true? Problem 1.4. Let (G, ρ) be a Polish metric space which is dense-in-itself. Let µ be a diffused non-σ quasi-finite Borel measure defined in (G, ρ). Do there exist a metric ρ 1 and a group operation ⊙ in G such that the following three conditions (j) The class of Borel measurable subsets of G generated by the metric ρ 1 coincides with the class of Borel measurable subsets of the same space generated by the metric ρ, (jj) (G, ρ 1 , ⊙) is a non-locally compact Polish group and (jjj) µ is a left(right or two-sided) invariant quasi-finite Borel measure in (G, ρ 1 , ⊙) hold true ?
In [4], the author uses methods of the theory ultrafilters to present a modified proof that a locally compact group with a countable basis has a left invariant and right invariant Haar measure. The author first shows that the topological space (β 1 X; τ 1 ) consisting of all ultrafilters on a non-empty set X is homeomorphic to the topological space (β 2 X; τ 2 ) of all nonzero multiplicative functions in the first dual space ℓ * ∞ (X) (Theorem 3.8). By using this result the author proves the existence of the infinitely additive left invariant measure λ on compact sets of the locally compact Hausdorf topological group G (Theorem 7.1). Starting from this point, the author introduces the notion of ν-measurable subsets in G where ν is an outer measure in G induced by the λ and open sets in G, and proves the existence of a left invariant Haar measure by the scheme presented in [7]. Notice that his proof essentially uses the axiom of choice. Several examples of the Haar measure are presented. It is underlined by Example 9.7 that G = R k with k = n 2 −n Problem 1.5([4],Question 9.8) What are the real k-dimensional manifolds with at least two different Lie group structures that have the same Haar measure?
The rest of the present manuscript is the following. In Section 2 we introduce a certain approach for an equipment of an arbitrary set of the cardinality of the continuum by structures of various(compact, locally compact or non-locally compact) Polish groups with two-sided(left or right) invariant Borel measures. Problems 1.2-1.5 will be stutied also in this section.
In Section 3 we consider a question asking whether an arbitrary diffused Borel probability measure in a Polish space without isolated points can be considered as a Haar measure.

Equipment of an arbitrary set with cardinality of the continuum with structures of Polish groups and Haar's measures
This section we begin by description of a certain useful approach which allows us to equip sets with cardinality of the continuum with structures of Polish groups and Haar's measures. Since next theorem (as we will see later) has many interesting applications, we give its proof in detail.
Theorem 2.1 Let X be a set with cardinality of the continuum and (G, ⊙, ρ) be a Polish group.
Further, let f : G → X be a one-to-one mapping. We set for x, y ∈ X. Then the following conditions hold true: is a locally compact Polish group then so is (X, ⊙ f , ρ f ); (vii) If (G, ⊙, ρ) a non-locally compact Polish group then so is (X, ⊙ f , ρ f ); (viii) If (G, ⊙, ρ) is a locally compact or compact Polish group and λ is a left(or right or twosided ) invariant Haar measure in (G, ⊙, ρ), then λ f also is a left(or right or two-sided ) invariant Haar (ix) If (G, ⊙, ρ) is a non-locally compact Polish group and λ is a left(or right or two-sided) invariant quasi-finite 3 Borel measure in (G, ⊙, ρ), then λ f also is a left(or right or two-sided) invariant quasi-finite Borel measure in

Proof. Proof of the item (i).
Closure Associativity . For all x, y and z in X, we have Identity element. Let e be an identity element of G. Setting e f := f (e) ∈ X, for x ∈ X we have The latter relations means that e f is the identity element of X. Inverse element. If a ∈ G then we denote its inverse element by a −1 we have to choose such neighbourhoods U X (x, r 1 ) and U X (y, r 2 ) of elements x and y respectively that ( We have . It is obvious to check the validity of the following equalities

Proof of the item (ii).
Since (G, ⊙) is an abelian Polish group, for x, y ∈ G f we have Take into account this fact and the associativity property of the group operation ⊙ f , we get that the condition Indeed assume the contrary and let x * be an isolated point of G f . The latter relation means that for We get the contradiction and the validity of the item (iv) is proved.
Proof of the item (v). We have to prove that if a family of open sets (U * i ) i∈I whose union covers the space G f then there is its subfamily whose union also covers the same space. Let consider a family of sets (f −1 (U * i )) i∈I . Since it is the family of open sets whose union covers the space G and G is a compact space, we claim that there is a finite subfamily has a compact neighbourhood U . Now it is obvious that the set f (U ) will be a compact neighbourhood of the point x * . Since x * ∈ G f was taken arbitrary the validity of the item (vi) is proved.
Proof of the item (vii). Since (G, ⊙, ρ) is no locally compact there is a point x 0 which has no a compact neighbourhood. Now if we consider a point f (x 0 ), we observe that it has no a compact neighbourhood. Indeed, if assume the contrary and U is a compact neighbourhood of the point f (x 0 ) then f −1 (U ) also will be a compact neighbourhood of the point x 0 and we get the contradiction. This ends the proof of the item (vii).

Proof of the item (viii).
Proof of the diffusivity of the measure λ f . Since λ vanishes on all singletons, we have for each x ∈ G f ; Proof of the left(or right or two-sided ) invariance of the measure λ f . If (G, ⊙, ρ) is a locally compact or compact Polish group and λ is a left(or right or two-sided ) invariant Haar measure in (G, ⊙, ρ), then λ f also will be a left(or right or two-sided ) invariant Haar measure is Borel σ-algebra of G f generated by the metric ρ f and λ f is defined by
Proof of the outer regularity of the measure λ f . Let take any set Proof of the inner regularity of the measure λ f . Let take any set Proof of the finiteness of the measure λ f on all compact subsets. Let take any compact set F ⊆ G f . Since f −1 (F ) is compact in G and the measure λ is finite on every compact set we get λ f (F ) = λ(f −1 (F )) < ∞.
Proof of the item (ix). The proof of this item can be obtained by the scheme used in the proof of the item (viii).
Below we consider some examples which employ the constructions described by Theorem 2.1.

International Journal of Advanced Research in Mathematics Vol. 5 13
Note that λ f defined by will be Haar measure in (−c, c), where λ denotes a linear Lebesgue measure in R. 1+e y for y ∈ R. It is well known(see, [5], Eq. 35, p. 5) that the relativistic law of adding velocities has the following By Theorem 2.1 we know that λ f is Haar measure in (0, +∞). Since we deduce that Note that Haar measure space (G, ·, ρ G , ν) constructed in [4](see p.54) coincides with Haar mea- We have Volume 5 and for x, y ∈ (−c, c). Then we get a new example of Haar measure space ( R f , + f , ρ f , λ f ). Note that the Haar measure

Remark 2.2
Let M be a topological space. A homeomorphism ϕ : U → V of an open set U ⊆ M onto an open set V ⊆ R d will be called a local coordinate chart (or just 'a chart') and U is then a coordinate neighbourhood (or 'a coordinate patch') in M .
A C ∞ differentiable structure, or smooth structure, on M is a collection of coordinate charts ϕ α : (ii) any two charts are 'compatible': for every α, β the change of local coordinates (iii) the collection of charts ϕ α is maximal with respect to the property (ii): if a chart ϕ of M is compatible with all ϕ α then ϕ is included in the collection.
A topological space equipped with a C ∞ differential structure is called a real smooth manifold. Then d is called the dimension of M , d = dimM .
Recall, that a Lie group is a set G with two structures: G is a group and G is a real smooth manifold. These structures agree in the following sense: multiplication and inversion are smooth maps.
In [4](see, Example 9.7, p. 64), it is shown that G = R k with k = n 2 −n 2 has two different Lie group structure and the Lebesgue measure in R k is Haar measure on both Lie groups. Further the author asks(cf. [4], Question 9.8) what are real k-dimensional manifolds with at least two different Lie group structures that have the same Haar measure.
The next example answers positively to Maleki's above mentioned question for each k > 2.

Remark 2.3
Notice that Example 2.5 covers Example 9.7 from [4]. Indeed, it is obvious that for n > 2, measure space has Lie group structure which differs from standard Lie group structure of R n because group operations ′′ + ′′ and ′′ + ′′ f , as were showed in Example 2.5, are different. Furthermore the Lebesgue measure λ n (in R n ) is Haar measure on both Lie groups.

Now let us consider
∞} as a vector space with usual addition operation " + ". If we equip ℓ 2 with standard metric ρ ℓ 2 defined by for (x k ) k∈ N , (y k ) k∈ N ∈ ℓ 2 , then (ℓ 2 , ′′ + ′′ , ρ ℓ 2 ) stands an example of a non-locally compact Polish group. Here naturally arise a question asking whether there exists a metric ρ in ℓ 2 such that ( ℓ 2 , "+", ρ ) stands an example of a locally compact σ-compact Polish group. An affirmative answer to this question is containing in the following example.

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Example 2.6 Let consider R and ℓ 2 as vector spaces over the group of all rational numbers Q. Let (a i ) i∈I and (b i ) i∈I be Hamel bases in R and ℓ 2 , respectively. For x ∈ R \ {0}, there exists a unique sequence of non-zero rational numbers (q . Note that f : R → ℓ 2 is one-to-one linear transformation. Let then we will obtain which means that a group operation + f coincides with usual addition operation " + ". Let ρ be defined by the formula By Theorem 2.1 we know that ( R f , + f , ρ f ), equivalently (ℓ 2 , +, ρ f ), is a locally compact noncompact Polish group which is isomorphic to the Polish group ( R, +, | · |).

Remark 2.4
Let (G, ρ, +) be an abelian Polish group. We say that G is one-dimensional group w.r.t. metric ρ if for each n ∈ N and for each family of different elements (a k ) 1≤k≤n there exists a permutation h of {1, 2, · · · , n} such that a h(k+1) ).

Example 2.7
Let consider R ∞ and R as vector spaces over the group of all rational numbers Q. Let (a i ) i∈I and (b i ) i∈I be Hamel bases in R ∞ and R, respectively. For x ∈ R ∞ \ {(0, 0, · · · )}, there exists a unique sequence of non-zero rational numbers (q )} and f (0, 0, · · · ) = 0. Note that f : R ∞ → R is one-to-one linear transformation.
We set and for x, y ∈ G. By Theorem 2.1 we know that (G, ⊙ φ , ρ φ ) is a compact Polish group without isolated points which is Borel isomorphic to the compact Polish group (G 2 , ⊙ 2 , ρ 2 ) and a measure λ φ , defined by is a two-sided invariant Haar measure in G.

Remark 3.1
In the proof of Theorem 3.1, if under (G 2 , ρ 2 , ⊙ 2 ) we take an abelian compact Polish group without isolated points and with a two-sided invariant Haar measure λ then the group (G, ρ φ , ⊙ φ ) will be a compact abelian Polish group without isolated points. Similar phenomena is valid in the case when (G 2 , ρ 2 , ⊙ 2 ) is a non-abelian compact Polish group without isolated points.
The solution of Problem 1.3 is contained in the following statement.
Theorem 3.2 Let (G, ρ) be a Polish metric space which is dense-in-itself. Let µ be a diffused σ-finite non-finite Borel measure defined in (G, ρ). Then there exist a metric ρ φ , a group operation ⊙ φ in G and the Borel measure µ ⋆ in G such that the following conditions (i) The class of Borel measurable subsets of G generated by the metric ρ φ coincides with the class of Borel measurable subsets of the same space generated by the metric ρ, (ii) (G, ρ φ , ⊙ φ ) is a non-compact locally compact Polish group, (iii) The measures µ ⋆ and µ are equivalent, and (iv) µ ⋆ is a left (right or two-sided) invariant σ-finite non-finite Haar measure in (G, ρ φ , ⊙ φ ) hold true.
Proof. Let (G 2 , ρ 2 , ⊙ 2 ) be a non-compact locally compact Polish group which is dense-in-itself with two-sided invariant σ-finite non-finite Haar measure λ 2 (for example, the real axis R with Lebesgue measure ). Let (X k ) k∈N be a partition of the G 2 into Borel measurable subsets such that 0 < λ 2 (X (2) k ) < +∞ for k ∈ N .
We set Similarly, let (Y k ) k∈N be a partition of the G into Borel measurable subsets such that 0 < µ(Y k ) < +∞ for k ∈ N . We set for Y ∈ B(G).
We have to show that the measure µ ⋆ is a two-sided invariant measure in G. Indeed, for h 1 , h 2 ∈ G and X ∈ B(G), we have Volume 5