Constructions and Properties of Quasi Sigma-Algebra in Topological Measure Space

Abstract

The topological views of a measure space provide deep insights. In this paper, the sigma-set algebraic structure is extended in a Hausdorff topological space based on the locally compactable neighborhood systems without considering strictly (metrized) Borel variety. The null extension gives rise to a quasi sigma-semiring based on sigma-neighborhoods, which are rectifiable in view of Dieudonné measure in n-space. The concepts of symmetric signed measure, uniformly pushforward measure, and its interval-valued Lebesgue variety within a topological measure space are introduced. The symmetric signed measure preserves the total ordering on the real line; however, the collapse of symmetry admits Dieudonné measure within the topological space. The locally constant measures in compact supports in sigma-neighborhood systems are invariant under topological deformation retraction in a simply connected space where the sequence of deformation retractions induces a strongly convergent sequence of measures. Moreover, the extended sigma-structures in an automorphic and isomorphic topological space preserve the properties of uniformly pushforward measure. The Haar-measurable group algebraic structures equivalent to additive integer groups arise under the locally constant and signed measures as long as the topological space is non-compact and the null-extended sigma-neighborhood system admits compact groups. The comparative analyses of the proposed concepts with respect to existing results are outlined.

1. Introduction

The concepts of $\sigma -$algebra (including Boolean algebra) and topology play a distinct role in the development of abstract measure theory. In general, the regular Borel measures are formulated considering a Hausdorff compact topological space involving the intersection number of a collection of open sets [1]. On the other hand, the $\sigma -$algebra of measurable sets is based on a set of limit points of Cauchy sequences preserved within the algebra of sets [2]. As a consequence, the Caratheodory extension theorem can be reformulated in view of $\mu -Cauchy$ sequences of sets in the respective measure space [2]. Note that the measures on a $\sigma -$ideal of the corresponding $\sigma -$field may vanish in a subspace topology containing the respective topological basis [1]. The concept of measurable cardinal numbers was first introduced by Ulam in 1930 [3]. Let $c$ be the cardinal number of a set $X$. The cardinal number $c$ is said to be measurable if the real-valued measure $\mu (.)$ is countably additive and such measure can be defined on every subset of $X$ [4]. The further developments of abstract measure theory in view of topology are highly attributed to Alexandroff (also known as Alexandrov) [5]. The topological notion of $\tau -additivity$ was introduced by Alexandroff in a series of seminal papers. Moreover, various pivotal constructions and concepts of abstract measure theory were introduced by Rohlin and Marczewski [6,7].

In general, the Borel measure in a measure space is formulated by the real-valued $\sigma -additive$ functions on Borel sets, where the functions are finite-valued in nature [8,9]. The Borel measure in a metrized topological space needs the sets of supports from closed balls $\{\overline{B(x_i,r>0)}:i\in \mathsf{\Lambda }\}$, where $\mathsf{\Lambda }$ denotes the index set and the measure is always non-zero within the supports. The reason is that the support sets are not considered to be in the Baire meager category. On the contrary, the measures with no sets of support also exist. For example, the Dieudonné measure is a class of Borel measure requiring no sets of support within the corresponding measure space [3]. It is important to note that the Borel measures and Baire measures in a topological space have distinctive properties. If we consider the algebra of Borel sets in a topological space, then a signed Borel measure is essentially a signed measure defined on the corresponding Borel sets in the topological space [10]. On the other hand, the signed Baire measure is formulated based on the $\sigma -$algebra generated by Baire sets in a topological space. According to Bourbaki, the measures can be constructed as a set of continuous linear functionals on a locally compact topological space; however, such an approach may not always yield richer properties [8]. The motivation of this research and the contributions made in this paper are presented in the following sections (Section 1.1 and Section 1.2, respectively).

1.1. Motivation

Let the space $X\ne \varphi$ be equipped with a $\sigma -$algebra denoted by $\sigma (X)$. It is well known that if $(X,\sigma (X),\mu )$ is a measurable space under measure $\mu (.)$, then the invertible endomorphism of the respective measure space preserves the left Haar measure of locally compact groups under translations [11]. On the other hand, let us consider a Borel space $Br(X)$ and the sigma-algebra over a measure space $X$, which is denoted as $(X,Br(X))$. It is known that, in a general Borel space $(X,Br(X))$, the finite positive measure $\mu :(X,Br(X))\to [0,+\infty )$ can be equipped with a set of supports given as $A\in Br(X),\mu (A)>0$, considering that $Br(X)\cong \sigma (X)$. Note that, in general, the finite and positive measures with regularity consider a (Borel) metric space (i.e., without resorting to a more general underlying topology which may not be strictly Borel). Interestingly, the sets of supports for probabilistic Borel measures are not well defined in the topological spaces in view of the topological concept of separability [3]. In 1939, it was shown by Dieudonné that the closed disjoint sets in inseparable topological spaces admit sets of support with full measure. Thus, there is an interplay between the existence of sets of support and separability in a topological measure space. This motivates a broad question: what are the interrelationships among the supports of a signed (and finite) symmetric measure in a Hausdorff topological measure space in view of quasi $\sigma -$algebra. This is to emphasize that we avoid the formation of topological hemicompactness and $\sigma -$compactness within the topological measure space under consideration. Furthermore, the relevant questions are: (1) is it possible to extend $\sigma -$structures in such topological measure spaces while relaxing the strict Borel variety, (2) how can the concept of symmetric signed measure be formulated and what are the properties of symmetric signed measures within such spaces in view of (topological) automorphism and homeomorphism, (3) how do the concepts of uniform measures, locally constant measures, and the uniformly pushforward measures behave in the presence of sets of support in a topological measure space containing quasi $\sigma -$algebra, and (4) what is the relationship to Dieudonné measure in this form and can it be admissible. These questions are investigated in relative detail in view of topology and abstract measure theory. The elements of group algebraic structures are employed as necessary to understand the related algebraic properties of the proposed constructions.

1.2. Contributions

The main contributions made in this paper can be summarized as follows: This paper extends $\sigma -$structures in view of topology to form a quasi $\sigma -$semiring preserving a set of supports in a Hausdorff topological measure space (i.e., the topological measure space is separable and not strictly metrized Borel). The extension avoids the formation of a sequence of topological hemicompact subspaces, and it relaxes the condition of formation of topologically $\sigma -$compact subspaces by allowing separation of compactable local neighborhoods within the topological measure space. Moreover, we do not consider the formation of $\mu -Cauchy$ sequences of measurable sets within the quasi $\sigma -$semiring. The concepts of symmetric signed measure, uniformly pushforward measure, and Lebesgue variety (LV) are introduced in this paper, and respective properties are analyzed in detail in view of topological measure theory. We selectively applied the concept of fuzzy measure with necessary refinements in the proposed uniform and constant measure by incorporating the equality of measures. It is shown that if the symmetry of the signed measure function is not maintained within the extended $\sigma -set$ over the entire space $X$, then the measure space degenerates into the Dieudonné variety in the corresponding separable (i.e., Hausdorff) neighborhoods in the topological space. Moreover, the Haar-measurable compact measure groups are formed under the locally constant and signed measures if certain conditions are maintained. In this paper, we are not emphasizing the translation invariance of Haar measure.

The remainder of the paper is organized as follows: The preliminary concepts and classical results are presented in Section 2. The definitions are presented in Section 3. The detailed analysis is presented in Section 4 as main results. Section 5 presents the discussions outlining the distinctive properties of the proposed concepts and topological constructions. Finally, Section 6 concludes the paper.

2. Preliminaries

The preliminary concepts and classical results are presented in two subsections as follows: the first section (Section 2.1) presents a set of properties related to Borel measures, and the second section (Section 2.2) presents the results related to topological views of measure spaces as well as rectifiability. In this paper the sets of real numbers and integers are denoted as $R$ and $Z$, respectively.

2.1. Borel Charges

The concepts of Borel measures in a metric space embody the classical results in measure theory. It is important to note that the signed Borel charge is defined on either the algebra of Borel sets or on the $\sigma -$algebra generated by Borel sets. The definition of Borel (signed) charge is given as follows [10]:

Definition 1

(outer regular charge). Let $B_X$ be the $\sigma -$algebra or algebra of measurable Borel sets in a topological space and let the measure $\mu (.)$ be a charge. The function $\mu (.)$ is an outer regular charge if $\forall A\in B_X,\mu (A)=\inf \{\mu (E):E\in B_X,A\subset E\}$, where $E=E^o$.

Evidently, the preservation of finite measure with Borel regularity requires compact sets. Recall that a compact set contains all of its limit points. An interesting case arises when a sequence of measurable real-valued functions does not converge, which results in the Hahn theorem as follows [10]:

Theorem 1.

If$(X,\sum )$is a measure space and${\langle f_n\rangle }_{n=1}^{+\infty }$is a sequence of measurable real-valued functions, then the set$E$of convergence points of${\langle f_n\rangle }_{n=1}^{+\infty }$is$\sum -$measurable.

Furthermore, it is possible to formulate a function $f:(X,\sum )\to (Y,d)$, where $(Y,d)$ is a separable metric space. However, the measurability of such a function is not always guaranteed and it needs a condition to be satisfied in terms of limit points [10].

2.2. Measure Spaces and Rectifiability

The interactions between topology and abstract measure theory have enriched the respective mathematical fields. In this section, a set of definitions and classical results are presented in view of topology and abstract measure theory. Suppose ${\mu }_{Borel}:Br(X)\to [0,+\infty )$ is a finite Borel measure in a Borel measure space in $X$, which is given as $(Br(X),{\mu }_{Borel})$. The support of the Borel measure in a measure space is defined as follows [12]:

Definition 2

(support to measure). Let $S\subset X$ be a subspace such that $S\ne \varphi$. The set $S$ is a support if the Borel measure admits the following two properties: (1) ${\mu }_{Borel}(X\backslash S)=0$ and (2) $C\subset X,[(C=\overline{C})\wedge ({\mu }_{Borel}(X\backslash C)=0)]\Rightarrow [C\subseteq S]$.

Note that the support set $S\subset X$ is said to be a quasiconvex set if every subset of $S\subset X$ contains at least one point from $S^o$ (i.e., in its interior) [13]. Moreover, a real-valued function $f(.)$ on a convex subspace is called quasiconcave if its negative, $-f(.)$, is quasiconvex. (Recall that a real-valued quasiconvex function $f(.)$ is defined on a convex set or interval $S$, where the set $\{x\in S:\forall r\in R,f(x)<r\}$ is convex.) It is interesting to note that the measure is vanishing almost everywhere outside of the set of supports within a Borel measure space. The notion of doubling measure in a metric (Borel) measure space is defined as follows [14]:

Definition 3

(doubling measure). In a Borel measure space $(Br(X),{\mu }_{Borel})$, there exists a constant $c\in (0,+\infty )$ such that $\forall x\in X$ the measure admits the ordering relation given as ${\mu }_{Borel}(B(x,2r))\le c{\mu }_{Borel}(B(x,r))$, where $0<r<\infty$.

The doubling measure is a finite and positive Borel measure, which can also be extended as ${\mu }_{Borel}:R^{d>1}\to [0,+\infty )$ in the $d-$dimensional real space. It results in the following theorem [15]:

Theorem 2.

If$d\ge 2$is the dimension of a real-measure space$R^d$, then there is a doubling Borel measure${\mu }_{Borel}:R^d\to [0,+\infty )$and a rectifiable function$f:[0,1]\to R^d$such that the measure admits${\mu }_{Borel}(f[0,1])>0$.

There is a close interrelationship between the concepts of general topology and measure theory. The concept of countably compact measure in a finite Borel measure space is defined as follows [16]:

Definition 4

(countably compact measure). A finite measure ${\mu }_{Borel}(.)$ is defined to be countably compact if it is inner regular with respect to the countably compact subsets of $Br(X)$.

Note that often such countably compact measure is also called a compact measure for simplicity. The compactness of a subspace has a relation to the rectifiable function in a measure space, which is presented as follows [17]:

Theorem 3.

A set$S$generated by a function$f:[0,1]\to X$is compact if the function$f(.)$is rectifiable and the set$S$is connected.

Proof. 

Let the function $f:[0,1]\to X$ be rectifiable such that if $(S\subseteq X)=f([0,1])$, then $S={\displaystyle \underset{i}{\cup }{g}_i([0,1])}$, where $\{g_i:[0,1]\to S,i\in Z^{+}\}$ is a set of continuous functions. This results in the property that ${\mu }_{Borel}(S)\in (0,+\infty )$ under the finite measure, where ${\mu }_{Borel}(X\backslash S)=0$ is identically vanishing. If the set $S\subseteq X$ is a (topologically) connected subspace, then $f:[0,1]\to X$ is a continuous function in $S$ admitting finite Borel measure. Moreover, every open cover of the set $S$ has subcover in $X$. Thus, the connected set $S\subseteq X$ generated by a rectifiable and continuous function is a compact topological subspace. □

Observe that the rectifiable function $f:[0,1]\to X$ generating a compact and connected subspace is continuous. It is important to note that, as a consequence, if $H^1$ is a Hausdorff measure in $d=1$, then the measure preserves the condition given by $H^1(S)<\infty$. Moreover, the following proposition relates the concept of basis in a topological space and the concept of supports of a Borel measure [3]:

Proposition 1.

If$(X,{\tau }_X)$is a first-countable topological space, then the support$\overline{B(x\in X,r>0)}$in$(Br(X),{\mu }_{Borel})$is well defined.

In other words, a separable metric topological space always admits a set of supports of the Borel measure within the space. It is important to note that, in general, measures are constructed based on Hausdorff spaces. However, the measures can also be formulated in non-Hausdorff spaces [18]. In such a construction of topological measure space the underlying space is considered to be a $T_0$ topological space.

3. Topological Quasi $\sigma -$ Algebra

The algebraic structure of a $\sigma -set$ and neighborhood systems in a topological measure space can be enriched with additional properties through the topological extensions without resorting to the strictly metrized Borel variety. For example, the concept of neighborhood systems in a space is presented in an axiomatic as well as a generalized form incorporating the algebraic properties of reflexivity and symmetry [19]. In this section, a set of definitions related to the extended $\sigma -set$ and an associated $\sigma -neighborhood$ system in a Hausdorff topological space is presented. We introduce the concept of symmetric measure in a topological measure space $(X,{\tau }_X)$, where an open neighborhood of closed $\left\{x_i\right\}\in {\tau }_X$ is denoted as $N_i$. However, the open neighborhood of a point $x_i\in X$ within $(X,{\tau }_X)$ is denoted as $U_X(x_i)$. The sets of real numbers and integers are denoted as $R$ and $Z$, respectively. The index set is denoted as $\mathsf{\Lambda }\subseteq Z^{+}$, and the isomorphism between $A$ and $B$ is algebraically denoted as $A{\cong }_{isom}B$.

Recall that a $\sigma -set$ denoted as $A\subset X$ is a sigma-algebraic structure generated on a sigma-semiring $S$ in a space $X$ if $A={\displaystyle \underset{i}{\cup }{B}_i}$ and $B_i\cap B_k=\varphi$, where $i\ne k$. The definition of an extended $\sigma -set$ is a topological extension, which is presented as follows:

Definition 5

(extended $\sigma -set$). Let $(X,{\tau }_X)$ be a Hausdorff first-countable topological space. If $S_H=\left\{\right\{x_i\}\in {\tau }_X:i\in \mathsf{\Lambda }\}$ is a subspace, then $\sigma (S_H)\subset {\tau }_X$ is an extended $\sigma -set$ generated by $S_H$ if the following properties are maintained:

$$\begin{array}{l}\sigma (S_H)=\{N_i\subset X:x_i\in N_i\}\cup \left\{\varphi \right\},\\ \forall i,k\in \mathsf{\Lambda },[i\ne k]\Rightarrow [\overline{N_i}\cap \overline{N_k}=\varphi ].\end{array}$$

(1)

Note that the extended $\sigma -set$ in a topological space $(X,{\tau }_X)$ is a countable and null-extended set of neighborhoods of discrete subspace $S_H$. Moreover, if $(X,{\tau }_X)$ is a compact space, then $\sigma (S_H)\backslash \left\{\varphi \right\}$ is countably finite in a first-countable topological space. If we take a subspace $S_{H<i,k>}\subset S_H$ such that $i<k$, then we can generate a $\sigma -neighborhood$ system denoted as $N_{<i,k>}$ in $(X,{\tau }_X)$, which is defined as follows:

Definition 6

($\sigma -neighborhood$ system). Let $(X,{\tau }_X)$ be a Hausdorff topological space where $S_{H<i,k>}\subset S_H$ is a topological discrete subspace. The corresponding $N_{<i,k>}$ in $(X,{\tau }_X)$ is a $\sigma -neighborhood$ system if the following conditions are preserved by the algebraic structure:

$$\begin{array}{l}\forall \left\{\right\{x_i\},\{x_k\left\}\right\}\subset S_{H<i,k>},(x_i\in N_i)\cap (x_k\in N_k)\ne \varphi ,\\ \exists N_{<i>}\subset N_i,N_{<k>}\subset N_k,(\{x_i\}\subset N_{<i>})\cap (\{x_k\}\subset N_{<k>})=\varphi ,\\ N_{<i,k>}={\displaystyle \underset{a\in [i,k]}{\cup }\left\{N_{<a>}\right\}}.\end{array}$$

(2)

It is important to note that the $\sigma -neighborhood$ system is defined considering the principle of first-countable neighborhood basis in the Hausdorff topological measure space (i.e., not from the existing extended $\sigma -set$).

Remark 1.

It can be observed that$\{N_{<a>}:i\le a\le k\}$is a general$\sigma -set$(i.e., not extended) and it has a quasi$\sigma -semiring$algebraic structure in a topological subspace, denoted as${\sigma }_{quasi}(A\subset X)$, where$N_{<i,k>}\subset {\tau }_A$. The reason for establishing a quasi$\sigma -semiring$is that$\forall N_{<i>}\in {\sigma }_{quasi}(A),\left({\displaystyle \underset{a\in \mathsf{\Lambda }}{\cup }(C_a\subset A)}\right)\subseteq N_{<i>}$, where$\forall a\exists b\in \mathsf{\Lambda },C_a\cap C_b\ne \varphi$without preserving the disjoint decomposition property of a general$\sigma -semiring$. Moreover, the empty set$\varphi$is not in$N_{<i,k>}\subset {\tau }_A$.

The general $\sigma -set$ structure given by $\{N_{<a>}:i\le a\le k\}$ in a topological measure space can be null extended to form a null-extended $\sigma -neighborhood$ system, which is defined as follows:

Definition 7

(null-extended $\sigma -neighborhood$ system). Let $N_{<i,k>}$ be a $\sigma -neighborhood$ in a topological space $(X,{\tau }_X)$. The corresponding null-extended $\sigma -neighborhood$ is defined as $N_{<i,k>}^{\varphi }=N_{<i,k>}\cup \left\{\varphi \right\}$.

It is important to note that if $S_{<u,v>}^{\varphi }=\{N_{<i,k>}^{\varphi }:1\le i\le u,1\le k\le v\}$ is a family of null-extended $\sigma -neighborhood$ systems in $(X,{\tau }_X)$, then it follows that $\forall N_{<i,k>}^{\varphi },N_{<m,n>}^{\varphi }\in S_{<u,v>}^{\varphi }$ and the following implication holds: $[(i\ne m)\wedge (k\ne n)]\Rightarrow [N_{<i,k>}^{\varphi }\cap N_{<m,n>}^{\varphi }=\{\varphi \}]$. As a result, the structure $N_{<i,k>}^{\varphi }$ is not qualified to be a $\sigma -semiring$ even after null extension because $\varphi \ne \left\{\varphi \right\}$. However, we can introduce the concept of uniform and locally constant measures in a topological measure space, which is defined as follows:

Definition 8

(uniform and locally constant measure). Let ${\mu }_{\sigma }:S_{<i,k>}^{\varphi }\to R$ be a real-valued signed measure. The measure is said to be a uniform a.e. measure under continuous $f:(X,{\tau }_X)\to (X,{\tau }_X)$ if $[f^{-1}(B)\subseteq A]\Rightarrow [{\mu }_{\sigma }(\overline{A})={\mu }_{\sigma }\overline{(f^{-1}(B))}]$, where $A=A^o$ and $B=B^o$. The measure is a locally constant measure if $\forall \left\{x_a\right\}\subset A,{\mu }_{\sigma }(\overline{A})={\mu }_{\sigma }(\{x_a\})$.

The uniform a.e. measure requires topological continuity of the function $f:(X,{\tau }_X)\to (X,{\tau }_X)$, and the concept of locally constant measure is a topological variety of a.e. measure in the neighborhood systems. It is important to note that if we restrict the function such that $f:(X,{\tau }_X)\to (R,{\tau }_R)$, then the function is a measurable variety. Two different varieties of examples are presented as follows to illustrate the concept:

Example 1.

An example can be derived from the concept of simple (measurable) function in the standard representation form given as$\lambda :X\to R$. Suppose the measure evaluates${\mu }_{\sigma }(\overline{N_{<i>}})=r$in the null-extended$\sigma -neighborhood$system and the measure$\omega :S_{<i,k>}^{\varphi }\to R$is an indicator function such that$\forall \left\{x_a\right\}\subset \overline{N_{<i>}}\subset X,\omega (\{x_a\})=1$and$\forall \left\{x_b\right\}\subset X\backslash \overline{N_{<i>}},\omega (\{x_b\})=0$. Thus, we can compute the simple representation of measure in standard form as$\lambda (x_a)={\displaystyle \sum _{n\in [i,k]}{c}_n}\omega (\{x_a\}\subset \overline{N_{<n>}})=c_i\in R$. As a result, if we fix$\forall x_a\in \overline{N_{<i>}},{\mu }_{\sigma }(\{x_a\})=\lambda (x_a)$and$c_i=r$, then we can conclude that the locally constant measure is attained as${\mu }_{\sigma }(\overline{N_{<i>}})={\mu }_{\sigma }(\{x_a\})$.

Example 2.

This example is derived without employing the indicator function and by applying the concept of support in a first-countable topological space presented earlier in Proposition 1 (without resorting to Borel variety here). Once again, we consider that$\forall \left\{x_a\right\}\subset \overline{N_{<i>}}\subset X,{\mu }_{\sigma }(\{x_a\})=r\in R$and$r\ne 0$, providing support to the measure in the respective neighborhood of the first-countable topological measure space under consideration. Suppose we compute the full measure in the null-extended$\sigma -neighborhood$system as${\mu }_{\sigma }(\overline{N_{<i>}})=\epsilon (\beta -n^{-1})\ne 0$, where$n\in Z^{+}$and$\epsilon ,\beta \in R$. This further leads to$\underset{n\to +\infty }{\lim }[{\mu }_{\sigma }(\overline{N_{<i>}})/{\mu }_{\sigma }(\left\{x_a\right\})]=1$if$\epsilon \beta =r$. As a result, we can conclude that${\mu }_{\sigma }(\overline{N_{<i>}})={\mu }_{\sigma }(\{x_a\})$as$n\to +\infty$in$N_{<i>}\subseteq {\displaystyle \underset{n\in \mathsf{\Lambda }}{\cup }{U}_X(x_n)}$.

It is important to note that the uniform and locally constant measure is a restrictive variety of fuzzy measure [20]. In other words, the uniform and locally constant measure preserves the equivalence relation of equality between the measures in the fuzzy measure formulation.

Definition 9

(symmetric measure). The measure ${\mu }_{\sigma }$ is a symmetric measure if it preserves the following conditions:

$$\begin{array}{l}\forall N_{<i,k>}^{\varphi }\in S_{<u,v>}^{\varphi },\exists [-r_{ik},r_{ik}]\subset R,\\ \{\sup {\mu }_{\sigma }(N_{<i,k>}^{\varphi })\}\cup \{\inf {\mu }_{\sigma }(N_{<i,k>}^{\varphi })\}=\{-r_{ik},r_{ik}\}.\end{array}$$

(3)

Remark 2.

The symmetric measure${\mu }_{\sigma }$maintains the consistency of measure because${\mu }_{\sigma }(\varphi )=0$and${\mu }_{\sigma }(N_{<i,k>}^{\varphi })={\mu }_{\sigma }(N_{<i,k>})={\displaystyle \sum _{a\in [i,k]}{\mu }_{\sigma }(\overline{N_{<a>}})}$. At this point, it is not emphasized whether the computation of symmetric measure is a sum of point measures or an interval-valued measure. This distinction has resulted in the following definition about the interval-valued measure known as Lebesgue variety (LV):

Definition 10

(Lebesgue variety). A symmetric measure ${\mu }_{\sigma }:S_{<i,k>}^{\varphi }\to R$ is said to be Lebesgue variety (LV) if ${\mu }_{\sigma }(N_{<a>})\cong (\{0\}\subset I_a\subset R)$ and is denoted as $LV[I_a]$.

Note that the symmetric $LV[I_a]$ attains standard Lebesgue measure in $I_a\subset R$. For example, if $I_a=[-r_a,r_a]$, then the corresponding standard Lebesgue measure is $L_m(I_a)=2r_a$. Similarly, we can compute the symmetric LV measure as ${\mu }_{\sigma }(N_{<i,k>}^{\varphi })={\displaystyle \underset{a\in [i,k]}{\cup }{\mu }_{\sigma }(\overline{N_{<a>}},N_{<a>}\in N_{<i,k>}^{\varphi })}$.

4. Main Results

The set of analytical results presented in this section are related to three fundamental characteristics in the field: (1) measure-theoretic properties of extended $\sigma -structures$ in the topological measure space, (2) analysis of behavior of measures under symmetry and under collapsed symmetry conditions (i.e., when the measure does not retain the symmetry property), and (3) properties of interactions of group algebraic structures, Haar measurability, and rectifiability of a function in the topological measure spaces. We present the main results in two parts. Section 4.1 presents the properties related to abstract measure theory. Next, Section 4.2 presents the topological and measurable group algebraic properties in detail.

4.1. Properties of Measures

The algebraic linear order-preserving property of the symmetric and signed measure is presented in the following theorem:

Theorem 4.

The symmetric and signed measure${\mu }_{\sigma }:S_{<u,v>}^{\varphi }\to R$preserves the total order in$R$if${\mu }_{\sigma }(\{x_i\})\ne {\mu }_{\sigma }(\{x_k\})$is a constant measure in every null-extended$\sigma -neighborhood$system in$S_{H<i,k>}$.

Proof. 

Let $(X,{\tau }_X)$ be a first-countable Hausdorff topological space and $S_{H<i,k>}\subset {\tau }_X$ be a subspace equipped with symmetric and signed measure ${\mu }_{\sigma }:S_{<u,v>}^{\varphi }\to R$ such that $[i\ne k]\Rightarrow [{\mu }_{\sigma }(\left\{x_i\right\})\ne {\mu }_{\sigma }(\left\{x_k\right\})]$ in the topological measure space. If we consider that $N_{<i,k>}^{\varphi }\in S_{<u,v>}^{\varphi }$, then it is true that $\forall N_{<a>},N_{<b>}\in N_{<i,k>}$ the symmetric and signed measure ${\mu }_{\sigma }:S_{<u,v>}^{\varphi }\to R$ maintains the condition given by ${\mu }_{\sigma }(\{x_a\}\subset N_{<a>})\ne {\mu }_{\sigma }(\{x_b\}\subset N_{<b>})$ if $a\ne b$. Moreover, if the symmetric and signed measure is a constant measure in a $\sigma -neighborhood$ system, then we conclude that $\forall N_{<a>}\in N_{<i,k>},{\mu }_{\sigma }(\overline{N_{<a>}})={\mu }_{\sigma }(\{x_a\})=r_a$. Suppose we consider that $r_a<0$. Now consider $\exists N_{<b>}\in N_{<i,k>}$ such that ${\mu }_{\sigma }(\overline{N_{<b>}})=(r_b>0)$. Thus, it results in a total ordering $r_a<0<r_b$ in $R$ induced by ${\mu }_{\sigma }:(A\subset N_{<i,k>}^{\varphi })\to R$, where $A=\{N_{<a>},N_{<b>},\varphi \}$, which is a null-extended subspace. Inductively, the symmetric and signed measure ${\mu }_{\sigma }:(N_{<i,k>}^{\varphi }\in S_{<u,v>}^{\varphi })\to R$ preserves total order in $[-r_{ik},r_{ik}]\subset R$, where $\{\sup {\mu }_{\sigma }(N_{<i,k>}^{\varphi })\}\cup \{\inf {\mu }_{\sigma }(N_{<i,k>}^{\varphi })\}=\{-r_{ik},r_{ik}\}$. Hence, the symmetric and signed measure ${\mu }_{\sigma }:S_{<u,v>}^{\varphi }\to R$ maintains total order in $R$ for every null-extended $\sigma -neighborhood$ system in $S_{H<i,k>}$ in $(X,{\tau }_X)$. □

The following proposition shows that we can admit Dieudonné measure in a separable and extended $\sigma -set$ under the degeneration of symmetry of the measure function (i.e., collapsed symmetry condition), along with additional criteria:

Proposition 2.

The set $S_H$is a support of symmetric signed measure${\mu }_{\sigma }:S_{<u,v>}^{\varphi }\to R$in a Hausdorff topological space$(X,{\tau }_X)$. Moreover, the measure${\mu }_{\sigma }:S_{<u,v>}^{\varphi }\to R$degenerates into the Dieudonné measure if the following two conditions are satisfied:$\{\sup {\mu }_{\sigma }(N_{<i,k>}^{\varphi })\}\cup \{\inf {\mu }_{\sigma }(N_{<i,k>}^{\varphi })\}=\{r_{ik}=1\}$and${\mu }_{\sigma }({\tau }_X\backslash \sigma (S_{H=X}))=0$, where$\sigma (S_{H=X})$represents the null-extended$\sigma -set$over$X$.

Proof. 

It is a natural consequence if we maintain the condition that $\forall N_i\in \sigma (S_H),{\mu }_{\sigma }(\{x_i\}\subset N_i)\ne 0$. Moreover, if we collapse the symmetry of measure ${\mu }_{\sigma }:S_{<u,v>}^{\varphi }\to R$ while preserving the support of $\forall N_i\in \sigma (S_{H=X})$ such that $\{\sup {\mu }_{\sigma }(N_{<i,k>}^{\varphi })\}\cup \{\inf {\mu }_{\sigma }(N_{<i,k>}^{\varphi })\}=\{r_{ik}=1\}$, then we can further restrict the measure function such that $\forall N_i\in \sigma (S_{H=X}),{\mu }_{\sigma }(\overline{N_i})=1$ and ${\mu }_{\sigma }({\tau }_X\backslash \sigma (S_{H=X}))=0$, which results in the formation of Dieudonné measure in a separable topological measure space through the degeneration of symmetry of measure. □

Remark 3.

In Section 2 (Theorem 2), it is mentioned that a rectifiable function has non-zero Borel measure; this concept can be adapted in this case in a modified form in view of Dieudonné measure. It is important to note that if$F_{X(i)}=\{f_{i,k}:[0,1]\to N_i\in \sigma (S_{H=X}),k\in \mathsf{\Lambda }\}$is a set of continuous functions, then it induces the condition that every connected and compact$\overline{N_i}$is rectifiable by${\displaystyle \underset{k\in [1,+\infty ]}{\cup }{f}_{i,k}\in F_{X(i)}}$, admitting the Dieudonné measure in the topological space with$\dim (X)=n$if the symmetry of measure is collapsed or degenerated in$\{0,1\}$.

The topological property of automorphism between first-countable spaces and the measure functions has an interesting interplay if the measure function is uniform within the automorphic spaces. The topologically automorphic spaces retain the equality of measures under function composition, which is presented in the following theorem:

Theorem 5.

If$f:S_{<u,v>}^{\varphi }\to S_{<u,v>}^{\varphi }$is an automorphism in$(X,{\tau }_X)$, then there exists an uniformly symmetric and signed measure${\mu }_{\sigma }:S_{<u,v>}^{\varphi }\to R$such that$({\mu }_{\sigma }\circ f^{-1})={\mu }_{\sigma }$.

Proof. 

Let $(X,{\tau }_X)$ be a first-countable Hausdorff topological space equipped with a set of null-extended $\sigma -neighborhood$ systems $S_{<u,v>}^{\varphi }$. If $f:S_{<u,v>}^{\varphi }\to S_{<u,v>}^{\varphi }$ is an automorphism, then $f(.)$ is a bijection such that $(f\circ f^{-1})(N_{<i,k>}^{\varphi })=I{d}_{<i,k>}$ and $(f^{-1}\circ f)(N_{<m,n>}^{\varphi })=I{d}_{<m,n>}$ are respective identity functions in the respective topological subspaces. Suppose there is a symmetric and signed measure ${\mu }_{\sigma }:S_{<u,v>}^{\varphi }\to R$ preserving uniformity under $f(.)$. It results in the conclusion that $\forall N_{<a>}\in N_{<i,k>}^{\varphi },\exists N_{<b>}\in N_{<m,n>}^{\varphi }$ the uniform measure function maintains the equality property of real-valued measure under the automorphism as $({\mu }_{\sigma }\circ I{d}_{<i,k>})=({\mu }_{\sigma }\circ I{d}_{<m,n>})$. Moreover, the uniform measure function preserves the measure consistency in the subspace as ${\mu }_{\sigma }(\varphi \in (N_{<i,k>}^{\varphi }\cap N_{<m,n>}^{\varphi }))=0$. Hence, it is concluded that $({\mu }_{\sigma }\circ f^{-1})={\mu }_{\sigma }$ in the set of null-extended $\sigma -neighborhood$ systems $S_{<u,v>}^{\varphi }$ under automorphism. □

Corollary 1.

If the uniform symmetric and signed measure preserves$({\mu }_{\sigma }\circ f^{-1})={\mu }_{\sigma }$under automorphic$f:S_{<u,v>}^{\varphi }\to S_{<u,v>}^{\varphi }$, then it also preserves the extreme points as$\{\sup {\mu }_{\sigma }(N_{<i,k>}^{\varphi }),\inf {\mu }_{\sigma }(N_{<i,k>}^{\varphi })\}=\{\sup {\mu }_{\sigma }(N_{<m,n>}^{\varphi }),\inf {\mu }_{\sigma }(N_{<m,n>}^{\varphi })\}$, where$i\ne m$and$k\ne n$.

Interestingly, the LV property of a symmetric and signed measure at local subspaces enforces the preservation of the LV property globally. This observation is presented in the following theorem:

Theorem 6.

In a set of null-extended$\sigma -neighborhood$systems$S_{<u,v>}^{\varphi }$, the symmetric LV measure${\mu }_{\sigma }(N_{<i,k>}^{\varphi })$is$LV[I_{ik}]$.

Proof. 

The proof is relatively straightforward. According to its definition, it is true that the symmetric LV measure in a $S_{<u,v>}^{\varphi }$ in $(X,{\tau }_X)$ maintains the property that $\forall LV[I_a],\exists LV[I_b]$ such that $LV[I_a]\cap LV[I_b]\ne \varphi$. Moreover, the symmetric LV measure maintains the conditions given as $\left({\displaystyle \underset{a\in [i,k]}{\cup }LV[I_a]}\right)\subset R$ and $\left\{0\right\}\subset {\displaystyle \underset{a\in [i,k]}{\cap }LV[I_a]}$. Moreover, if every $I_a$ is compact, then ${\displaystyle \underset{a\in [i,k]}{\cup }LV[I_a]}$ is also compact and the standard Lebesgue measure of ${\mu }_{\sigma }(N_{<i,k>}^{\varphi })$ is $2r_{ik}>0$ under the symmetry. Hence, we can conclude that ${\mu }_{\sigma }(N_{<i,k>}^{\varphi })$ is $LV[I_{ik}]$. □

Remark 4.

If${\mu }_{\sigma }(N_{<i,k>}^{\varphi })$is$LV[I_{ik}]$in a$S_{<u,v>}^{\varphi }$, then$L_m\left({\displaystyle \underset{a\in [i,k]}{\cup }{I}_a}\right)<{\displaystyle \sum _{a\in [i,k]}{L}_m(I_a)}$.

The uniformness of the symmetric and signed measure in a topological space invites the concept of uniformly pushforward measure in two isomorphic topological spaces. This observation is presented in the following theorem:

Theorem 7.

If the$(S_H{|}_X\subset {\tau }_X){\cong }_{isom}(S_H{|}_Y\subset {\tau }_Y)$condition is maintained in the respective Hausdorff first-countable topological spaces$(X,{\tau }_X)$and$(Y,{\tau }_Y)$, then$({\mu }_{\sigma }\circ f^{-1})$is a uniformly pushforward measure, where$f:(X,{\tau }_X)\to (Y,{\tau }_Y)$is an isomorphism and${\mu }_{\sigma }:((S_{<u,v>}^{\varphi }{|}_X)\cup (S_{<w,z>}^{\varphi }{|}_Y))\to R$is uniform a.e.

Proof. 

Let $f:(X,{\tau }_X)\to (Y,{\tau }_Y)$ be an isomorphism between two Hausdorff and first-countable topological spaces. Suppose we consider subspaces $S_H{|}_X\subset {\tau }_X$ and $S_H{|}_Y\subset {\tau }_Y$ in the respective topological spaces such that $(S_H{|}_X\subset {\tau }_X){\cong }_{isom}(S_H{|}_Y\subset {\tau }_Y)$ under the continuous function $f:(X,{\tau }_X)\to (Y,{\tau }_Y)$. The isomorphism preserves the condition given as $\forall N_a\in \sigma (S_H{|}_X),\exists N_b\in \sigma (S_H{|}_Y)$ such that $f^{-1}(N_b)\subseteq N_a$ maintains the continuity of $f:(X,{\tau }_X)\to (Y,{\tau }_Y)$. Thus, if ${\mu }_{\sigma }:((S_{<u,v>}^{\varphi }{|}_X)\cup (S_{<w,z>}^{\varphi }{|}_Y))\to R$ is the symmetric measure with uniform a.e., then the uniform measure maintains the condition as ${\mu }_{\sigma }\overline{(f^{-1}(N_{<b>}))}={\mu }_{\sigma }(\overline{N_{<a>}})$. Hence, the composition $({\mu }_{\sigma }\circ f^{-1})$ is a uniformly pushforward measure in two isomorphic topological measure spaces. □

Remark 5.

The concept of uniformly pushforward measure between isomorphic topological spaces is stricter than the generalized pushforward measure. The reason is that the general pushforward measure is given by$({\mu }_{Borel}\circ f^{-1})(B\in \sigma (Y))={\mu }_{Borel}(A\in \sigma (X))$, where the function$f:(X,\sigma (X))\to (Y,\sigma (Y))$is a measurable map without enforcing topological continuity. Additionally, if the continuous function$f:(X,{\tau }_X)\to (Y,{\tau }_Y)$is a homeomorphism, then we can also induce a uniform pushforward measure in the homeomorphic topological measure spaces, although additional conditions are required.

4.2. Topological and Group Algebraic Properties

Recall from Section 2 (Theorem 3) that the interrelationship between topology and measure theory shows that a compact and connected set can be generated by a rectifiable continuous function. The interplay between the topological deformation retractions of a simply connected space and the symmetric measure is presented in the following theorem. It is shown that the order-inducing symmetric and locally constant measure retains the linear ordering property in the real numbers under deformation retractions.

Theorem 8.

If$(X,{\tau }_X)$is a simply connected topological space equipped with the symmetric and locally constant measure${\mu }_{\sigma }:S_{<u,v>}^{\varphi }\to R$, then$({\mu }_{\sigma }\circ \gamma )={\mu }_{\sigma }$, where$\gamma :N_{<i,k>}\to S_H$is a deformation retraction and the measure induces total ordering in$R$.

Proof. 

Let $(X,{\tau }_X)$ be a Hausdorff topological space equipped with the symmetric and locally constant measure ${\mu }_{\sigma }:S_{<u,v>}^{\varphi }\to R$. If the space is a simply connected variety, then there is a deformation retraction given by $\gamma :N_{<i,k>}\to S_H$ such that $\forall N_{<a>}\in N_{<i,k>},\gamma (\overline{N_{<a>}})=(\{x_a\}\subset S_H)$. Note that the set $\left\{x_a\right\}$ is closed in the Hausdorff space $(X,{\tau }_X)$. Moreover, if ${\mu }_{\sigma }:S_{<u,v>}^{\varphi }\to R$ is a measure within the space inducing a total ordering in $R$, then it is a surjective (constant) measure in $N_{<i,k>}^{\varphi }$ and we can conclude that ${\mu }_{\sigma }(\overline{N_{<a>}})={\mu }_{\sigma }(\{x_a\})$. Hence, it satisfies the condition given by $({\mu }_{\sigma }\circ \gamma )(\overline{N_{<a>}})={\mu }_{\sigma }(\overline{N_{<a>}})$. □

Example 3.

Let us consider a smooth and closed planar disk$D_2\subset R^2$such that it retains the homeomorphism given as$\mathrm{hom}(\partial D_2,S^1)$. As the topological space$(X=R^2,{\tau }_X)$is simply connected, we can conclude that$\exists (x_a,y_a)\in D_2^o$such that$\gamma (\partial D_2)=\{(x_a,y_a)\}$. However, if the measure${\mu }_{\sigma }:R^2\to R$is a locally constant measure in the$S_{<u,v>}^{\varphi }$structure in$(X=R^2,{\tau }_X)$, then we can infer that $({\mu }_{\sigma }\circ \gamma )(D_2)={\mu }_{\sigma }(\{(x_a,y_a)\})$.

Interestingly, if we relax the condition that the signed measure is a locally constant variety (i.e., considering ${\mu }_{\sigma }(\overline{N_{<a>}})\ne {\mu }_{\sigma }(\{x_a\})$) in $N_a\in \sigma (S_H)$, then a relationship with the corresponding LV measure under deformation retraction can be found, which is presented in the following proposition:

Proposition 3.

If the signed measure in$N_i\in \sigma (S_H)$is a$LV[I_i]$, then the corresponding measure under deformation retraction preserves the relation as$|{\mu }_{\sigma }(\overline{N_{<i>}})|=\epsilon |({\mu }_{\sigma }\circ \gamma )(\overline{N_{<i>}})|,\epsilon >0$, provided$\left\{x_i\right\}\subset N_{<i>}$is a support of${\mu }_{\sigma }(.)$and$|{\mu }_{\sigma }(N_{<i>}\backslash \left\{x_i\right\})|>0$is a.e. in$N_{<i>}$.

Remark 6.

Note that we have extended the concept of support sets to a closed point-set support in the structure$\sigma (S_H)$in the Hausdorff topological space.

It is observed that the Grothendieck sets in the algebra of sets admit a pointwise convergent measure sequence of bounded scalar-valued measures and the measure sequence is weakly convergent in the corresponding Banach space [21]. Thus, a similar question can be asked in the null-extended $\sigma -neighborhood$ system under finite measures. A set of new analytical properties emerges if we take a sequence of topological retractions in a null-extended $\sigma -neighborhood$ system. It is possible to attain two forms of convergence depending upon the natures of measure functions. This observation is presented in the following lemma:

Lemma 1.

If the function sequence${\langle {\gamma }_q\rangle }_{q=1}^n$is a sequence of retractions in a simply connected topological space such that$[s<t]\Rightarrow [{\gamma }_t(\overline{N_{<a>}})\subset {\gamma }_s(\overline{N_{<a>}})]$, then$\forall N_{<a>}\in N_{<i,k>}$the sequence of measures${\lim }_{n\to +\infty }[{\langle ({\mu }_{\sigma }\circ {\gamma }_q)(\overline{N_{<a>}})\rangle }_{q=1}^n]$is strongly convergent in$R$if the measure is a constant measure. Moreover, on the contrary, if the finite positive measure is not a constant variety, then${\lim }_{n\to +\infty }[{\langle ({\mu }_{\sigma }\circ {\gamma }_q)(\overline{N_{<a>}})\rangle }_{q=1}^n]\to [|{\mu }_{\sigma }(\{x_a\})|>0]$is also convergent.

Proof. 

The proof is an immediate consequence considering two cases: (1) the constant real-valued measure and (2) non-constant finite and positive measure. In the first case, the locally constant measure maintains the condition that $\forall n\in Z^{+},({\mu }_{\sigma }\circ {\gamma }_n)(\overline{N_{<a>}})={\mu }_{\sigma }(\overline{N_{<a>}})$, and if ${\mu }_{\sigma }(\overline{N_{<a>}})=r_a$, then $R\backslash \left\{r_a\right\}$ results in the separated subspaces in the real numbers indicating that ${\lim }_{n\to +\infty }[{\langle ({\mu }_{\sigma }\circ {\gamma }_q)(\overline{N_{<a>}})\rangle }_{q=1}^n]$ is strongly convergent in $R$. In the second case, if we consider that $\forall n,m\in Z^{+},[n>m]\Rightarrow [({\mu }_{\sigma }\circ {\gamma }_n)(\overline{N_{<a>}})>({\mu }_{\sigma }\circ {\gamma }_m)(\overline{N_{<a>}})]$, then it results in the following conditions:

$$\begin{array}{l}\forall n\in Z^{+},0<({\mu }_{\sigma }\circ {\gamma }_n)(\overline{N_{<a>}})<+\infty ,\\ [n\to +\infty ]\Rightarrow [({\mu }_{\sigma }\circ {\gamma }_n)(\overline{N_{<a>}})\to {\mu }_{\sigma }(\left\{x_a\right\})].\end{array}$$

(4)

Thus, in both cases the measure sequences under topological retractions of a simply connected space are convergent. This completes the proof. □

Example 4.

Suppose${\mu }_{\sigma }(.)$is a locally constant measure. In this case, if${\mu }_{\sigma }(\overline{N_{<a>}})=r_a$, then the measure sequence generated under retraction is${\langle r_n=r_a\rangle }_{n=1}^{+\infty }$, which is a strongly convergent sequence in the real numbers. On the other hand, if${\mu }_{\sigma }(.)$is not a locally constant finite measure and$({\mu }_{\sigma }\circ {\gamma }_n)(\overline{N_{<a>}})=r_n$, then$({\lim }_{n\to +\infty }{r}_n)\to l_a$, where${\mu }_{\sigma }(\{x_a\})=l_a$. Thus, the resulting measure sequence given by${\langle r_n\rangle }_{n=1}^{+\infty }$is bounded and convergent.

However, it further leads to a more interesting and insightful property as presented in the following corollary:

Corollary 2.

Suppose the measure${\mu }_{\sigma }:S_{<u,v>}^{\varphi }\to R$is a constant measure preserving the total order in$A\subset R$. If$N_{<i,k>}$is finitely countable up to$m\in [1,+\infty )$, then${\langle {\mu }_{\sigma }(\overline{N_{<a>}})\rangle }_{a=1}^m$is a bounded sequence of measures in$A\subset R$and the respective${\langle {\langle ({\mu }_{\sigma }\circ {\gamma }_q)(\overline{N_{<a>}})\rangle }_{q=1}^n\rangle }_{a=1}^m$is also a bounded sequence of sequences in$A\subset R$, where every subsequence is strongly convergent.

Suppose we consider a non-compact Hausdorff topological space $(X,{\tau }_X)$ and we prepare $S_{H=X}=\left\{\right\{x_i\}\in {\tau }_X:i\in \mathsf{\Lambda }\}$ such that the structure retains the properties of $\sigma (S_{H=X})$ within the entire $(X,{\tau }_X)$. As a result, the locally constant and signed measure ${\mu }_{\sigma }:\sigma (S_{H=X})\to (Z\subset R)$ generates an additive measure-group structure $G_{\sigma }=(\sigma (S_{H=X}),{\mu }_{\sigma },+)$, as presented in the following theorem:

Theorem 9.

If${\mu }_{\sigma }:\sigma (S_{H=X})\to (Z\subset R)$is a locally constant and signed measure in a non-compact$(X,{\tau }_X)$, then$G_{\sigma }=(\sigma (S_{H=X}),{\mu }_{\sigma },+)\cong (Z,+)$.

Proof. 

Let $(X,{\tau }_X)$ be a non-compact Hausdorff topological space equipped with a signed and locally constant measure given as ${\mu }_{\sigma }:\sigma (S_{H=X})\to (Z\subset R)$. This results in the property that $[i\ne k]\Rightarrow [{\mu }_{\sigma }(\overline{N_i})\ne {\mu }_{\sigma }(\overline{N_k})]$. Moreover, the measure maintains that ${\mu }_{\sigma }(\varphi \in \sigma (S_{H=X}))=0$ and ${\displaystyle \underset{i\in \mathsf{\Lambda }}{\cup }{\mu }_{\sigma }(\overline{N_i})}=Z$. Hence, we conclude that $G_{\sigma }=(\sigma (S_{H=X}),{\mu }_{\sigma },+)$ is a measure group and $G_{\sigma }=(\sigma (S_{H=X}),{\mu }_{\sigma },+)\cong (Z,+)$. □

Interestingly, the Haar measure in a set of compact topological groups $G_X=\{(\overline{N_i},{\ast }_i):N_i\in \sigma (S_{H=X})\}$ fails to retain the group structure of $(Z,+)$, where ${\ast }_i:\overline{(N_i}\times \overline{N_i})\to \overline{N_i}$ is closed. This observation is presented in the following lemma:

Lemma 2.

If we consider that$G_X=\{(\overline{N_i},{\ast }_i):N_i\in \sigma (S_{H=X})\}$is a set of compact topological groups in$(X,{\tau }_X)$such that${\ast }_i:\overline{(N_i}\times \overline{N_i})\to \overline{N_i}$is closed, then$G_X$is Haar-measurable by${\mu }_{\sigma }{|}_{Haar}:G_X\to Z^{+}$and it fails to attain$(Z,+)$.

Proof. 

It is straightforward to conclude that $(G_X,{\mu }_{\sigma }{|}_{Haar},+)$ is not $(Z,+)$ because ${\mu }_{\sigma }{|}_{Haar}(G_X)>0$ and $G_X$ is not null extended. □

Finally, we emphasize that the non-compactness of a topological space $(X,{\tau }_X)$ is an important condition to retain $G_{\sigma }=(\sigma (S_{H=X}),{\mu }_{\sigma },+)\cong (Z,+)$. Moreover, the Haar measurability is restricted in the compact topological groups placed in disjoint subspaces within the non-compact topological space $(X,{\tau }_X)$.

5. Discussions

In this section, we revisit some of the relevant concepts and contemporary works while outlining the distinctive properties of the proposed topological constructions presented in this paper. We comprehensively present a comparative study considering various existing concepts and results.

5.1. Relation to Fuzzy Measures

In general, the monotone convergence sequence theorem in a measure space considers the property that $\mu (E_n)\in R$ and $\mu (\underset{n\to +\infty }{\lim }{E}_n)=\underset{n\to +\infty }{\lim }\mu (E_n)$, where every set in the sequence is finitely measurable [22]. Moreover, in this case the measure space is considered to be metrizable, sigma-finite, and the sequence of measures of the sets is convergent (but not strongly convergent). Lusin’s theorem is a classical result in measure theory and it does not hold if the measure is non-additive in nature. It is shown that Lusin’s theorem is preserved in real-valued fuzzy Borel measures on a metric space if the weakly null-additivity condition is maintained [23]. It is important to note that in the topological vector space settings, the measure $\mu :F\to E,\mu (\varphi )=0$ is non-additive if the following implication holds: $A,B\in F,[A\subset B]\Rightarrow [\mu (A)\le \mu (B)]$, where $F$ is a $\sigma -$field over $X\ne \varphi$ and $E$ is a vector space endowed with a topology [24]. A non-additive measure is a fuzzy measure if it is continuous from above and from below. Evidently, the topological measure space proposed in this paper is not a complete fuzzy measure space. Note that the regularity of non-additive measures can be retained in the Hausdorff topological vector space [24]. Similarly, the regularity of measure can be preserved in the proposed topological measure space. Note that a standard (Borel) measure space does not fully admit the concepts of fuzzy measures. However, as a distinction, the topological constructions proposed in this paper incorporate the concepts of fuzzy measures within the topological measure space with the necessary refinements, and the underlying measure space is not necessarily based upon the topological space of Borel variety. The refinements allow the introduction of uniform a.e. measure and locally constant measure in a compactable sigma-neighborhood system within the underlying topological measure space. As a result, this leads to the formation of a strongly convergent sequence of measures under topological retractions in the measure space.

5.2. Covers and Probability Measures

Note that a strongly convergent sequence of measures depends on a stronger criterion as compared with the general convergence of a sequence of measures in a monotone sequence of sets. However, the measures in a sigma-neighborhood system of a topological measure space preserve the property of sigma-finiteness. The distinctive property of the sigma-finite measure in a sigma-neighborhood system is that the measurable sets in a sigma-neighborhood system do not cover the entire topological measure space. Thus, the proposed topological construction distinguishes between the local covering of a measure space and the (topological) covering of the entire space. As a result, there is no restriction that the topological measure space needs to be always compact. In a probability measure space, the Prohorov theorem invites the concept of relative compactness. A family of probability measures is called relatively compact if every sequence of elements in the family of probability measures contains a weakly convergent subsequence [25]. Evidently, the notion of relative compactness in a probability measure space is very different from the topological views of compactness in a topological measure space.

A complete and separable metric space can be equipped with probability measure and two such spaces can preserve isometry in terms of respective probability measures via mappings. It is shown that the equivalence classes generated under such isometry can be metrized by employing the Gromov-Prohorov metric, and the Cartesian product of two such metric measure spaces equipped with a binary operation gives rise to a commutative Polish semigroup [26]. Interestingly, if we consider the resulting equivalence classes as single points, then only the identity element of the semigroup admits probability measure within the metric measure spaces equipped with binary operation [26]. However, the proposed constructions of the topological measure space in this paper are of a more relaxed and generalized variety, not requiring any specific metric structure, binary operation, or Cartesian products between measure spaces. The additive (i.e., commutative) measure groups are formed within the sigma-neighborhood systems under the locally constant and signed measure. The necessary requirements are that the underlying topological measure space should be non-compact and the sigma-neighborhood system should be null extended, forming a quasi sigma-semiring structure. Moreover, in this case, every locally compact group in a sigma-neighborhood system within the topological measure space is Haar-measurable.

5.3. Relation to Rectifiability

In the case of determination of rectifiability of m-dimensional measures in an n-dimensional space, it is considered that the underlying measure space is Euclidean [27]. Moreover, it is considered that the Hausdorff measures in such spaces are affected by the degree of smoothness of functions on a differentiable submanifold structure. On the contrary, the topological constructions proposed in this paper do not require the space to be strictly (globally) Euclidean and the measures are independent of the degree of smoothness of functions. However, the similarity between the rectifiability in an m-dimensional Euclidean space and the proposed topological constructions is that both admit the concept of neighborhood retracts.

6. Conclusions

The sigma-set algebraic structure is null-extended in a Hausdorff topological space to form a quasi sigma-semiring, which is finitely measurable. The quasi sigma-semiring can be viewed as a countable collection of locally compactable neighborhood systems within the underlying topological measure space, which are rectifiable. The resulting topological measure space admits the concepts of symmetric signed measure, uniformly pushforward measure, locally constant measure, and its interval-valued Lebesgue variety. The formulation of symmetric signed measure preserves total ordering in the real numbers. However, the degeneration of symmetry of the signed measure admits Dieudonné measure within the corresponding topological measure space. The locally constant measures in compact supports in the sigma-neighborhood systems are invariant under topological retractions in a simply connected topological space, where the sequence of retractions induces a strongly convergent measure sequence. Moreover, the quasi sigma-semiring in the automorphic and isomorphic topological spaces admits uniformly pushforward measures. The Haar-measurable compact group algebraic structures arise under the locally constant and signed measures, which are equivalent to the additive integer groups provided that the topological measure space is a non-compact space and the respective sigma-neighborhood system is null extended.