On Measurable Separable-Normal Radon Measure Manifold

Transcription

1 Global Journal of Pure and Applied Mathematics. ISSN Volume 1, Number 1 (016), pp Research India Publications On Measurable Separable-Normal Radon Measure Manifold S. C. P. Halakatti Associate Professor, Department of Mathematics, Karnatak University, Dharwad , Karnataka, India. Soubhagya Baddi Research Scholar, Department of mathematics, Karnatak University, Dharwad , Karnataka, India. Abstract In this paper, we show that a non-empty set of composition of C measurable homeomorphisms and Radon measure-invariant functions induce a group structure on the set M = { M 1, M,., M n } of measurable separable-normal Radon measure manifolds where the measurable separable-normal property remains invariant under measurable homeomorphisms and Radon measureinvariant function. Keywords: Measurable separable-normal Radon measure manifold, Measurable homeomorphism and Radon measure-invariant map. Mathematics Subject Classification: 8-XX, 54-XX, 58-XX, 57N-XX. 1. Introduction The concept of Measure Manifold was introduced and the invariance of Radon measurable normality property on such measure manifold under C measurable homeomorphism and Radon measure-invariant transformation was investigated and proved by the author S. C. P. Halakatti [4], [7], [8]. In this paper, the invariance of separable-normal property P, μ R a. e. (where μ R is a Radon measure) on Radon measure manifold under C measurable homeomorphism and Radon measure-invariant map is investigated by S. C. P. Halakatti. Further, it is shown that a non-empty set of such C measurable homeomorphisms and Radon measure-invariant functions induce a group structure on M where the property P

2 1096 S. C. P. Halakatti and Soubhagya Baddi remains invariant. On these developed ideas we continue to study some more properties on measurable separable-normal Radon measure manifolds.. Preliminaries In this section, we consider some basic concepts: Definition.1: Radon Measure on (R n, τ, Σ, μ) [11]: A Radon measure on a topological measure space (R n, τ, Σ, μ) is a positive Borel measure μ : B [0, ] which is finite on compact subsets and is inner regular in the sense that for every Borel subset E (R n, τ, Σ, μ) we have (i) μ(e) = sup{ μ(k) : K E, K K} where K denote the family of all compact subsets. μ is outer regular on a family F of Borel subsets if for every E F (R n, τ, Σ, μ) we have, (ii) μ(e) = inf{ μ(o) : O E, O O} where O denote the family of all open subsets. Definition.: Measurable Homeomorphism [5]: Let (M, τ) be a second countable, Hausdorff topological space and (R n, τ, Σ, μ) be a measure space. Then the function φ U M (R n, τ, Σ, μ) is called measurable homeomorphism if (i) φ is bijective and bi continuous (ii) φ and φ 1 are measurable Definition.3: Dense subset of a topological space [10]: A subset Α of a topological space X is called dense in X if closure of Α is X i.e., Α = X. Definition.4: Separable topological space [10]: A topological space (R n, τ) is said to be separable if it has countable dense subsets. Inverse Function Theorem on Measure Manifold [5]: Let F (M, τ, Σ, μ) (M 1, τ 1, Σ 1, μ 1 ) be a C measurable homeomorphism and measure-invariant map of measure manifolds and suppose that F p : T p (M) T f(p) (M 1 ) is a linear isomorphism at some p of M. Then there exists a measure chart (U, φ) of p in M such that the restriction of F to (U, φ) is a diffeomorphism onto a measure chart (V, φ) of F(p) in M 1. This implies for every function F which is measurable homeomorphism and measure-invariant has a C F 1 : (M 1, τ 1, Σ 1, μ 1 ) (M, τ, Σ, μ) which is also measurable homeomorphism and measure-invariant.

3 On Measurable Separable-Normal Radon Measure Manifold 1097 Definition.5: Radon measure chart [7]: A measurable chart ((U,τ 1/U,Σ 1/U ), φ) of a measurable manifold (M,, τ 1, Σ 1 ) equipped with a Radon measure μ R1/U is called a Radon measure chart denoted by ((U, τ 1/U,Σ 1/U, μ R1/U ), φ) satisfying the following conditions: a) φ is measurable homeomorphism, if for every compact Borel subset V (R n, τ, Σ), φ 1 (V ) = (U, φ) (M, τ 1, Σ 1 ) is also Borel measurable. b) φ is Radon measure-invariant. that is, μ R1 (φ 1 (V )) = μ R (V ) where φ 1 (V ) = (U, φ) a Borel measurable chart which satisfies the Radon measure conditions: I. (i) For p V V Σ ; μ R (V ) < ; (ii) For any Borel compact subset V (R n, τ, Σ), μ R (V ) = sup{μ R (E i ); i I : E i V : E i compact and measurable} II. (i) For φ 1 (V ) Σ 1, where Σ 1 a σ-algebra induced on second countable Hausdorff topological space, μ R1 (φ 1 (V )) < ; where φ 1 (V ) = (U, φ) is a measurable chart (ii) For any Borel subset φ 1 (V ) (M, τ 1, Σ 1 ), μ R1 (φ 1 (V )) = sup{μ R1 (φ 1 (E i )) ; i I : φ 1 (E i ) φ 1 (V ) : φ 1 (E i ) compact and measurable}. Definition.6: Radon measure atlas [7]: By an R n -Radon measure atlas of class C k (k 1) on measurable manifold (M, τ 1, Σ 1 ), we mean a countable collection (A,τ 1/A,Σ 1 / A,μ R1/A ) of n-dimensional Radon measure charts ((U i,τ 1/Ui,Σ 1/Ui,μ R1/Ui ),φ i ) for all i N on (M, τ 1, Σ 1, μ R1 ) satisfying the following conditions: (a 1 ) ((U i, τ 1/Ui, Σ 1/Ui, μ R1/Ui ) i I = (M, τ 1, Σ 1, μ R1 ) i.e., the countable union of all Radon measure charts in (A,τ 1/A,Σ 1 / A,μ R1/A ) cover (M, τ 1, Σ 1, μ R1 ). (a ) For any pair of Radon measure charts ((U i,τ 1/Ui,Σ 1/Ui,μ R1/Ui ),φ i ) and ((U j,τ 1/Uj,Σ 1/Uj,μ R1/Uj ),φ j ) in (A,τ 1/A,Σ 1 / A,μ R1/A ), the transition maps φ i φ j 1 and φ j φ i 1 are: 1. differentiable maps of class C k (k 1) i.e., φ i φ j 1 : φ j (U i U j ) φ i (U i U j ) (R n, τ, Σ,μ R ) and φ j φ i 1 : φ i (U i U j ) φ j (U i U j ) (R n, τ, Σ,μ R ) are differentiable maps of class C k (k 1).. measurable Transition maps φ i φ j 1 and φ j φ i 1 are measurable functions if, a) any Borel subset K φ i (U i U j ) is measurable in (R n,τ,σ,μ R ) then, (φ i φ j 1 ) 1 (K) φ j (U i U j ) is also measurable. b) φ j φ i 1 is measurable if S φ j (U i U j ) is measurable in (R n,τ, Σ,μ R ), then (φ j φ i 1 ) 1 (S) φ i (U i U j )is also measurable.

4 1098 S. C. P. Halakatti and Soubhagya Baddi c) For any two Radon measure atlases (A 1,τ,Σ 1/A 1 1/ A,μ R1/A and 1 1) (A,τ,Σ 1/A 1/ A,μ R1/A we say that a mapping T: A 1 A is measurable if ), T 1 (E) is measurable for every Radon measurable chart A = (U,τ 1/U,Σ 1 / U,μ R1/U ) (A,τ,Σ 1/A 1/ A,μ R1/A and the mapping is Radon measure preserving if ) μ R1/A = μ R1/A satisfying the Radon measure conditions: 1 I. (i) p A A, μ R1/A < ; where A is a Borel compact subset of A. (A) (ii) μ R1/A (A) = sup{ μ /A (E i) ; i I : E i A: E i compact and measurable}. T 1 : A A 1 is Radon measure preserving transformation if μ R1/A (E) = μ R1/A (T 1 (E)) where A 1 ~A and μ R1/A = μ R1/A the Radon 1 1satisfying measure conditions: II. (i) T 1 (p) T 1 (A) A 1 ; μ R1/A 1(T 1 (A)) < ; (ii) μ R1/A 1(T 1 (A)) = sup{ μ /A (T 1 (E i )); i I: T 1 (E i )) T 1 (A): T 1 (E i ) compact and measurable} Then, we call T a Radon-measure preserving transformation. (a 4 ) If a measurable transformation T: A A preserves a Radon measure μ R1, then we say that μ R1 is T-invariant. If T is invariant and if both T and T 1 are measurable and Radon measure preserving then we call T an invertible Radon measure preserving transformation. Let A k m (M) denotes the set of all Radon measure atlases of class C k on (M, τ 1, Σ 1, μ R1 ). Two Radon measure atlases A 1 and A in A k m (M) are said to be equivalent if (A 1 A ) A k m (M). In order to have that A 1 A to be a member of A k m (M) we require that for every Radon measure chart (U i, φ i ) A 1 and (V j, φ j ) A, the set of φ i (U i V j ) and φ j (U i V j ) are Borel subsets in (R n 1,τ, Σ,μ R ) and maps φ i φ j and φ j φ 1 i are of class C k and are measurable. The relation introduced is an equivalence relation in A k m (M) and hence partitions A k m (M) into disjoint equivalence classes. Each of these equivalence classes forms a differentiable structure of class C k on (M, τ 1, Σ 1, μ R1 ). Definition.7: Radon measure manifold [7]: A measurable manifold (M, τ 1, Σ 1 ) equipped with a Radon measure μ R1 is called a Radon measure manifold (M, τ 1, Σ 1, μ R1 ). 3. Main Results The following results are introduced and developed by S. C. P. Halakatti and we carry our study on these results.

5 On Measurable Separable-Normal Radon Measure Manifold 1099 Definition 3.1: Measurable separable-normal Radon measure space: A Radon measure space (R n, τ, Σ, μ R ) is said to be measurable separable-normal, if for any two disjoint dense Borel subsets E i and E j which are Radon measurable there exists two Borel subsets A i and A j of (R n, τ, Σ, μ R ) such that E i A i, E j A j, A i A j = and μ R (A i ) > 0 and μ R (A j ) > 0. By using theorem 3.4 [7], the measurable separable-normal property can be studied on a Radon measure manifold (M, τ 1, Σ 1, μ R1 ) as follows: Theorem 3.1: Let (R n, τ, Σ, μ R ) be a measurable separable-normal Radon measure space and (M, τ 1, Σ 1, μ R1 ) be a Radon measure manifold. If there exists a C measurable homeomorphism and Radon measure-invariant map φ (M, τ 1, Σ 1, μ R1 ) (R n, τ, Σ, μ R ) then (M, τ 1, Σ 1, μ R1 ) is a measurable separablenormal Radon measure manifold. Proof: Suppose (R n, τ, Σ, μ R ) is a measurable separable-normal Radon measure space and (M, τ 1, Σ 1, μ R1 ) is a Radon measure manifold. If there exists a C measurable homeomorphism and Radon measure-invariant map φ (M, τ 1, Σ 1, μ R1 ) (R n, τ, Σ, μ R ) then we show that (M, τ 1, Σ 1, μ R1 ) is a measurable separable-normal Radon measure manifold by using theorem 3.4[7]. Since (R n, τ, Σ, μ R ) is measurable separable-normal with Radon measure μ R, then there exists two disjoint dense Borel subsets E i and E j K and two Borel subsets A i and A j G which are Radon measurable in (R n, τ, Σ, μ R ) such that E i A i and E j A j and E i = (R n, τ, Σ, μ R ) or E j = (R n, τ, Σ, μ R ) and A i A j =, also μ R (A i ) > 0 and μ R (A j ) > 0 where K is countable family of countable dense/compact Borel subsets of (R n, τ, Σ, μ R ) and G is the family of Borel subsets of (R n, τ, Σ, μ R ). On (R n, τ, Σ, μ R ), E i and E j are Radon measurable. Since E i and E j are compact Borel subsets of (R n, τ, Σ, μ R ), for every Borel cover E i n { i I P i } there exists a Borel sub cover { j=1 P ij } and for every Borel cover E j n { i I Q i } there exists a Borel sub cover { j=1 Q ij } such that P ij E i and Q ij E j such that: I. For E i A i (R n, τ, Σ, μ R ), (i) μ R (E i ) <, (ii) μ R (E i ) = sup{ μ R (P ij ) : j J, P ij E i, P ij K }, (iii) μ R (E i ) = inf{ μ R (A i ) : i I, A i E i, A i G } II. For E j A j (R n, τ, Σ, μ R ), (i) μ R (E j ) <, (ii) μ R (E j ) = sup{ μ R (Q ij ) : j J, Q ij E j, Q ij K }, (iii) μ R (E j ) = inf{ μ R (A j ) : j J, A j E j, A j G }.

6 1100 S. C. P. Halakatti and Soubhagya Baddi If φ (M, τ 1, Σ 1, μ R1 ) (R n, τ, Σ, μ R ) is a C measurable homeomorphism and Radon measure-invariant map, then by using theorem 3.4[7], for each E i, E j (R n, τ, Σ, μ R ), φ 1 (E i ) = K i, φ 1 (E j ) = K j which are dense /compact Borel subsets belonging to (M, τ 1, Σ 1, μ R1 ) and for each A i, A j (R n, τ, Σ, μ R ) φ 1 (A i ) = (U i, φ i ), φ 1 (A j ) = (U j, φ j ) are Radon measure charts belonging to (M, τ 1, Σ 1, μ R1 ) such that K i = (M, τ 1, Σ 1, μ R1 ) or K j = (M, τ 1, Σ 1, μ R1 ) and K i (U i, φ i ), K j (U j, φ j ), K i K j =, μ R1 (U i ) > 0, μ R1 (U j ) > 0 and K i, K j φ 1 (K) = F where F is a countable family of dense/compact Borel subsets and (U i, φ i ), (U j, φ j ) A where A is the family of Radon measure charts of (M, τ 1, Σ 1, μ R1 ), which is called Radon measure atlas of (M, τ 1, Σ 1, μ R1 ). On(M, τ 1, Σ 1, μ R1 ), K i and K j are Radon measurable. Since K i and K j are compact Borel subsets of (M, τ 1, Σ 1, μ R1 ), for every Borel cover K i { i I R i } there exists a n Borel sub cover { j=1 R ij } and for every Borel cover K j { i I S i } there exists a n Borel sub cover { S ij } such that R ij K i and S ij K j such that: j=1 III. For K i U i (M, τ 1, Σ 1, μ R1 ), (i) μ R1 (K i ) <, (ii) μ R1 (K i ) = sup{ μ R1 (R ij ) : j J, R ij K i, R ij F }, (iii) μ R1 (K i ) = inf{ μ R1 (U i ) : i I, U i K i, U i A } IV. For K j U j (M, τ 1, Σ 1, μ R1 ), (i) μ R1 (K j ) <, (ii) μ R1 (K j ) = sup{ μ R1 (S ij ) : j J, S ij K j, S ij F }, (iii) μ R1 (K j ) = inf{ μ R1 (U j ) : j J, U j K j, U j A }. Since φ is measurable homeomorphism and Radon measure-invariant, for each E i (R n, τ, Σ, μ R ) φ 1 (E i ) = K i (M, τ 1, Σ 1, μ R1 ) such that μ R1 (φ 1 (E i )) = μ R (E i ) μ R1 (K i ) = μ R (E i ) and for each E j (R n, τ, Σ, μ R ) φ 1 (E j ) = K j (M, τ 1, Σ 1, μ R1 ) such that μ R1 (φ 1 (E j )) = μ R (E j ) μ R1 (K j ) = μ R (E j ). This shows that the existence of C measurable homeomorphism and Radon measure-invariant map φ (M, τ 1, Σ 1, μ R1 ) (R n, τ, Σ, μ R ) generates a measurable separable-normal Radon measure manifold (M, τ 1, Σ 1, μ R1 ). Hence the following definition: Definition 3.: Measurable separable-normal Radon measure manifold: A Radon measure manifold (M, τ 1, Σ 1, μ R1 ) is said to be measurable separable-normal if for any two disjoint dense Borel subsets K i and K j which are Radon measurable

7 On Measurable Separable-Normal Radon Measure Manifold 1101 there exists two Radon measure charts (U i, φ i ) and (U j, φ j ) of (M, τ 1, Σ 1, μ R1 ) such that K i (U i, φ i ), K j (U j, φ j ) and (U i, φ i ) (U j, φ j ) =, μ R1 (U i ) > 0, μ R1 (U j ) > 0. Using the above theorem, The Inverse Function Theorem on Measure Manifolds [5] can be extended on Radon measure manifold as follows: Inverse Function Theorem on Radon measure manifolds: Let F (M 1, τ 1, Σ 1, μ R1 ) (M, τ, Σ, μ R ) be a C measurable homeomorphism and Radon measure-invariant map of Radon measure manifolds (M 1, τ 1, Σ 1, μ R1 ) and (M, τ, Σ, μ R ) where μ R1 and μ R are Radon measures on (M 1, τ 1, Σ 1, μ R1 ) and (M, τ, Σ, μ R ) respectively and suppose that F p : T p (M 1 ) T f(p) (M ) is a linear isomorphism at some point p of M 1. Then there exists a Radon measure chart (U, φ) of p in M 1 such that the restriction of F to (U, φ) is a diffeomorphism onto a Radon measure chart (V, φ) of F(p) in M. This implies for every function F which is measurable homeomorphism and Radon measure-invariant has a C map F 1 : (M, τ, Σ, μ R ) (M 1, τ 1, Σ 1, μ R1 ) which is also measurable homeomorphism and Radon measure-invariant. In the following example, S. C. P. Halakatti has shown that the set D of all dyadic rationals is Radon measurable on measurable separable Radon measure manifold (R 1, τ, Σ, μ R ). Example 3.1: The subset D of all dyadic rationals in [0,1] is Radon measurable on measurable separable Radon measure manifold (R 1, τ, Σ, μ R ). Solution: Let (R 1, τ, Σ, μ R ) be a measurable separable Radon measure manifold. Let D = { a b, where a is an integer and b is a natural number } i.e. D = { 0,..., 1 4, 1 3, 3 4, 1, 5 4, 3 3, 7 4, 1, 9 4, 5 3,..., 1 }. D is relatively small dense subset in [0,1] (R 1, τ, Σ, μ R ), i.e. D = [0,1] (R 1, τ, Σ, μ R ). Since the subset of all dyadic rationals is dense in compact set [0,1] (R 1, τ, Σ, μ R ), it admits Radon measure satisfying inner and outer regularity properties. We show that D has outer regular and inner regular measure μ R on D : (i) The outer regular measure μ R on D is as follows: μ R (D ) = lim inf d b { d b a c : d b in [0,1]} d d = 0 Therefore, the outer regular measure μ R on D is 0. (ii) The inner regular measure μ R on D is as follows: μ R (D ) = lim sup d b { d b a c : d b in [0,1]} d d

8 110 S. C. P. Halakatti and Soubhagya Baddi = lim d sup d b { d b a d c d c c b = lim d sup d b { d b a = lim d sup d b { a d : d b in [0,1]} d: d b in [0,1]} d: d b in [0,1]} = a b If a = 0, b = 1,, 3 then a b = 0 If a = 1, b = 1,, 3 then 1, 1, 1 3,... If a =, b = 1,, 3 then,, 3,... Now if a =, b = 1 then a b = 1 = 1 Therefore, the inner regular measure μ R on D is 1 i.e. μ R (D ) = 1. Hence, the Radon measure μ R which is a positive Borel measure is outer regular having the value zero and inner regular having the value 1 on the dense subset μ R (D ) = [0,1]. Note: In this paper, by a Radon measure manifold we mean a measurable separable-normal Radon measure manifold. Also, we denote the set of all C measurable homeomorphisms and Radon measure-invariant functions F 1, F,., F n by G = { F 1, F,., F n }. Remark: We observe that a measurable separable-normal property say P holds μ R a. e. on any dense Borel set A (M, τ, Σ, μ R ) if the set A = { K i (M, τ, Σ, μ R ): P(K i ) is true } has positive measure i.e. μ R (A) > 0. Suppose P does not hold μ R a. e. on the set A (M, τ, Σ, μ R ) then μ R (A) = 0. Then we identify A as a dark region of (M, τ, Σ, μ R ). Now by using Inverse Function Theorem on Radon measure manifold, S. C. P. Halakatti has shown the following result and we conduct study on it. Theorem 3.: Let (M 1, τ 1, Σ 1, μ R1 ) and (M, τ, Σ, μ R ) be Radon measure manifolds. If F (M 1, τ 1, Σ 1, μ R1 ) (M, τ, Σ, μ R ) is a C measurable homeomorphism and Radon measure-invariant map and if the measurable separable-normal property say P holds μ R a. e. on (M, τ, Σ, μ R ) then P holds μ R1 a. e. on (M 1, τ 1, Σ 1, μ R1 ). Proof: Let (M 1, τ 1, Σ 1, μ R1 ) and (M, τ, Σ, μ R ) be Radon measure manifolds of dimension n. Let P be a measurable separable-normal property holds μ R a. e. on the set A of dense Borel subsets K i in (M, τ, Σ, μ R ), where A = { K i (M, τ, Σ, μ R ) : P(K i ) is true } has positive measure i.e. μ R (A) > 0.

9 On Measurable Separable-Normal Radon Measure Manifold 1103 We show that P also holds μ R1 a. e. on (M 1, τ 1, Σ 1, μ R1 ). Since F (M 1, τ 1, Σ 1, μ R1 ) (M, τ, Σ, μ R ) is a C measurable homeomorphism and Radon measure-invariant map from (M 1, τ 1, Σ 1, μ R1 ) to (M, τ, Σ, μ R ) there exists F 1 : (M, τ, Σ, μ R ) (M 1, τ 1, Σ 1, μ R1 ). If measurable separable-normal property P holds μ R a. e. on the set A of dense Borel subsets K i in (M, τ, Σ, μ R ) where the set A = { K i (M, τ, Σ, μ R ) : P(K i ) is true } has positive measure i.e. μ R (A) > 0 on (M, τ, Σ, μ R ) then the property P also holds μ R1 a. e. on the set F 1 (A) of dense Borel subsets F 1 (K i ) in (M 1, τ 1, Σ 1, μ R1 ) where F 1 (A) = {F 1 (K i ) (M 1, τ 1, Σ 1, μ R1 ) : P( F 1 (K i )) is true } has positive measure i.e. μ R1 (F 1 (A)) > 0 on (M 1, τ 1, Σ 1, μ R1 ). Also since F is C measurable homeomorphism and Radon measure-invariant map, if P holds μ R1 a. e. on any dense set A (M 1, τ 1, Σ 1, μ R1 ) where the set A = { K i (M 1, τ 1, Σ 1, μ R1 ) : P(K i ) is true } has positive measure i.e. μ R1 (A) > 0 on (M 1, τ 1, Σ 1, μ R1 ), then the property P also holds μ R a. e. on any dense set F(A) (M, τ, Σ, μ R ) where F(A) = { F(K i ) (M, τ, Σ, μ R ) : P(F(K i )) is true} has positive measure i.e. μ R (F(A)) > 0 on (M, τ, Σ, μ R ). Therefore, if measurable separable-normal property P holds μ R a. e. A (M, τ, Σ, μ R ) then the property P also holds μ R1 a. e. F 1 (A) (M 1, τ 1, Σ 1, μ R1 ) under C measurable homeomorphism and Radon measure-invariant map F (M 1, τ 1, Σ 1, μ R1 ) (M, τ, Σ, μ R ). Theorem 3.3: Let (M 1, τ 1, Σ 1, μ R1 ), (M, τ, Σ, μ R ) and (M 3, τ 3, Σ 3, μ R3 ) be Radon measure manifolds. Let F 1 (M 1, τ 1, Σ 1, μ R1 ) (M, τ, Σ, μ R ) and F (M, τ, Σ, μ R ) (M 3, τ 3, Σ 3, μ R3 ) be C measurable homeomorphisms and Radon measure-invariant maps. Then if measurable separable-normal property say P holds μ R1 a. e. on A (M 1, τ 1, Σ 1, μ R1 ) then P also holds μ R3 a. e. on (M 3, τ 3, Σ 3, μ R3 ) under the composition mapping F F 1 (M 1, τ 1, Σ 1, μ R1 ) (M 3, τ 3, Σ 3, μ R3 ). Proof: Let (M 1, τ 1, Σ 1, μ R1 ), (M, τ, Σ, μ R ) and (M 3, τ 3, Σ 3, μ R3 ) be Radon measure manifolds. Let F 1 (M 1, τ 1, Σ 1, μ R1 ) (M, τ, Σ, μ R ) and F (M, τ, Σ, μ R ) (M 3, τ 3, Σ 3, μ R3 ) be C measurable homeomorphisms and Radon measure-invariant maps. We show that, if measurable separable-normal property P holds μ R1 a. e. on (M 1, τ 1, Σ 1, μ R1 ) then P holds μ R3 a. e. on (M 3, τ 3, Σ 3, μ R3 ) under the composition mapping F F 1 (M 1, τ 1, Σ 1, μ R1 ) (M 3, τ 3, Σ 3, μ R3 ). From theorem 3., P holds μ R1 a. e. on any dense set A (M 1, τ 1, Σ 1, μ R1 ) where A = { K i (M 1, τ 1, Σ 1, μ R1 ) : P(K i ) is true } has positive measure i.e. μ R1 (A) > 0, then P also holds μ R a. e. on any dense set F 1 (A) (M, τ, Σ, μ R ) where F 1 (A) = {

10 1104 S. C. P. Halakatti and Soubhagya Baddi F 1 (K i ) (M, τ, Σ, μ R ) : P(F 1 (K i )) is true } has positive measure i.e. μ R (F(A)) > 0 under the C measurable homeomorphism and Radon measure-invariant map F 1 (M 1, τ 1, Σ 1, μ R1 ) (M, τ, Σ, μ R ). Similarly, if the property P holds μ R a. e. on any dense set F 1 (A) (M, τ, Σ, μ R ) where F 1 (A) = { F 1 (K i ) (M, τ, Σ, μ R ) : P(F 1 (K i )) is true } has positive measure i.e. μ R (F 1 (A)) > 0, then the property P also holds μ R3 a. e. on any dense set F (F 1 (A)) (M 3, τ 3, Σ 3, μ R3 ) where F (F 1 (A)) = { F (F 1 (K i )) (M 3, τ 3, Σ 3, μ R3 ) : F (F 1 (K i )) is true } has positive measure i.e. μ R3 (F (F 1 (A))) > 0 under C measurable homeomorphism and Radon measure-invariant map F (M, τ, Σ, μ R ) (M 3, τ 3, Σ 3, μ R3 ). Since F (F 1 (A)) = F F 1 (A), we have F F 1 (A) = { F F 1 (K i ) (M 3, τ 3, Σ 3, μ R3 ) : P(F F 1 (K i )) is true} has positive measure i.e. μ R3 (F F 1 (A)) > 0 and also since F 1 and F are C measurable homeomorphisms and Radon measureinvariant maps, F F 1 (M 1, τ 1, Σ 1, μ R1 ) (M 3, τ 3, Σ 3, μ R3 ) is also C measurable homeomorphism and Radon measure-invariant map. Also, for every F F 1 (M 1, τ 1, Σ 1, μ R1 ) (M 3, τ 3, Σ 3, μ R3 ) there exists an inverse map (F F 1 ) 1 : (M 3, τ 3, Σ 3, μ R3 ) (M 1, τ 1, Σ 1, μ R1 ) such that if P holds μ R3 a. e. on any dense Borel set F F 1 (A) (M 3, τ 3, Σ 3, μ R3 ) where F F 1 (A) = { F F 1 (K i ) (M 3, τ 3, Σ 3, μ R3 ) : P(F F 1 (K i )) is true} has positive measure i.e. μ R3 (F F 1 (A)) > 0, then the property P also holds μ R1 a. e. on any dense Borel set (F F 1 ) 1 (A) (M 1, τ 1, Σ 1, μ R1 ) where (F F 1 ) 1 (A) = { (F F 1 ) 1 (K i ) (M 1, τ 1, Σ 1, μ R1 ) : P (F F 1 ) 1 (K i ) is true } has positive measure i.e. μ R1 (F F 1 ) 1 (A) > 0. Therefore, if P holds μ R1 a. e. A (M 1, τ 1, Σ 1, μ R1 ) then P also holds μ R3 a. e. F F 1 (A) (M 3, τ 3, Σ 3, μ R3 ) under C measurable homeomorphism and Radon measure-invariant map F F 1 (M 1, τ 1, Σ 1, μ R1 ) (M 3, τ 3, Σ 3, μ R3 ). Hence the composition map F F 1 preserves measurable separable-normal property μ a. e.. In these two results, we have shown that, if measurable separable-normal property P holds μ R1 a. e. on (M 1, τ 1, Σ 1, μ R1 ) then it holds μ R a. e. on (M, τ, Σ, μ R ) under C measurable homeomorphism and Radon measure-invariant function F 1 and if P holds μ R a. e. on (M, τ, Σ, μ R ) then it holds μ R3 a. e. on (M 3, τ 3, Σ 3, μ R3 ) under C measurable homeomorphism and Radon measure-invariant function F and also if measurable separable-normal property P holds μ R1 a. e. on (M 1, τ 1, Σ 1, μ R1 ) then it also holds μ R3 a. e. on (M 3, τ 3, Σ 3, μ R3 ) under the composition F F 1. By continuing this process, one can show that any (M i, τ i, Σ i, μ Ri ) can be related to any (M j, τ j, Σ j, μ Rj ) by composition of two C measurable homeomorphisms and Radon measure-invariant functions F i F j denoted as M i R M j.

11 On Measurable Separable-Normal Radon Measure Manifold 1105 Thus, if there exists F 1 M 1 M, F M M 3,..., F n M n 1 M n, then the composition of functions F 1, F,., F n induces a group structure (G, ) on the nonempty set of Radon measure manifolds (M 1, M,., M n ) = M. That is, if (G, ) = { F 1, F,., F n } is a non-empty set of composition of F 1, F,., F n and if M = { M 1, M,., M n } is a set of Radon measure manifolds then, S. C. P. Halakatti has shown that (G, ) induces a group structure on M. The following result is introduced and proved by S. C. P. Halakatti and we conduct a study on it. Theorem 3.4: Let M = { M 1, M,., M n } be a non-empty set of Radon measure manifolds and G = { F 1, F,., F n } be the set of C measurable homeomorphisms and Radon measure-invariant functions on M. Then (G, ) induces a group structure on M under the composition of C measurable homeomorphism and Radon measureinvariant functions F i and F j, 1 i, j n. Proof: Let M = { M 1, M,., M n } be the set of Radon measure manifolds and let G = { F 1, F,., F n } be the set of C measurable homeomorphisms and Radon measure-invariant functions on M. In theorem 3.3, it is shown that if F 1 : M 1 M and F : M M 3 are C measurable homeomorphisms and Radon measure-invariant functions, then F F 1 : M 1 M 3 and F 1 F : M 3 M 1 are also C measurable homeomorphisms and Radon measure-invariant functions. We show that (G, ) induces a group structure on M. (i) If F i, F j (G, ) then F i F j (G, ) [By theorem 3.3] (ii) If F i, F j, F l (G, ) then (F i F j ) F l = F i (F j F l ) Let F i F j = F R and F j F l = F S Then, F R F l = F i F S [By theorem 3.3] (iii) For every F i (G, ) there exists an inverse map F 1 i (G, ) such that F i F 1 i = F 1 i F i = id. [By theorem 3., since both F i and F 1 i are C measurable homeomorphism and Radon measure-invariant functions] (iv) For any F i (G, ) there exists an identity map id : F i F i (G, ) such that id F i = F i id = F i holds, where id (G, ). [By theorem 3.3] Therefore, (G, ) = { F 1, F,. F i, F 1 i, id,., F n } induces a group structure on M under the composition of C measurable homeomorphisms and Radon measureinvariant functions. If M is a non-empty set of maximally connected measure manifolds, S. C. P. Halakatti has observed that the group structure (G, ) on M = {M 1, M,., M n } generates a Network structure (M, G, ) of order 1.

12 1106 S. C. P. Halakatti and Soubhagya Baddi Conclusion The measure manifold (M, τ, Σ, μ R ) satisfying the separable-normal property P is invariant under the composition of finite number of measurable homeomorphisms and Radon measure-invariant functions induces a group structure (G, ) on M. If it is maximally connected measure manifold then (M, G, ) generates a Network structure of order 1. References [1] Bogachev V. I., 1998, "Measures on Topological Spaces," Journal of Mathematical Sciences, Volume 91, No. 4. [] Bogachev V. I., 006, "Measure Theory", Volume II, Springer. [3] Dorlas T. C., 010, "Remainder Notes for the Course on Measure on Topological Spaces", Dublin Institute for Advanced Studies, School of theoretical Physics, 10, Dublin 4, Ireland. [4] S. C. P. Halakatti and H. G. Haloli,, 014,"Introducing the Concept of Measure Manifold (M,τ 1, Σ 1, μ 1 )", International Journal of Scientific Research, Journal of Mathematics (IOSR-JM),10, Issue 3, Ver. II, [5] S. C. P. Halakatti, Akshata Kengangutti and Soubhagya Baddi, 015,"Generating A Measure Manifold", International Journal of Mathematical Archieve, (IJMA), Volume 6, Issue 3, [6] S. C. P. Halakatti and H. G. Haloli, 014,"Extended Topological Properties on Measure Manifold", International Journal of Mathematical Archieve, (IJMA), Volume 05, Issue 3, [7] S. C. P. Halakatti and Soubhagya Baddi,, 015,"Radon Measure on Compact Measurable Manifold with Some Measurable properties", International Journal of Mathematical Archieve, (IJMA), Volume 6, Issue 7, [8] S. C. P. Halakatti and Soubhagya Baddi, 015, "Radon Measure on Compact Topological Measurable Space", International Journal of Scientific Research, Journal of Mathematics (IOSR-JM), Volume 11, Issue, Ver. III, [9] Hunter John K, 011, "Measure Theory, Lecture notes", University of California at Davis, -11, 0, 33. [10] Patty C. W., "Foundations of Topology", Jones and Bartlett Publishers, Inc., First Indian Edition. [11] William Arveson, 1996, "Notes on Measure and Integration in Locally Compact Spaces", Department of Mathematics, University of California, Berkeley. [1] Fremlin D. H., (007), "Measure Theory", Volume 4, University of Essex.