Convergence and Connectedness on Complete Measure Manifold

Convergence and Connectedness on Complete Measure Manifold

S. C. P. Halakatti* Department of Mathematics, Karnatak University Dharwad,

Karnataka, India

Abstract – In this paper we study different modes of

H. G. Haloli

Research scholar, Department of Mathematics, Karnatak University Dharwad,

Karnataka, India

A measure 1 U on measurable chart ((U, 1 U , 1 U), )

convergence and introduce some aspects of connectedness like, locally path connected 1 – a.e., interconnected 1 – a.e., maximal connected 1 – a.e. on a complete measure manifold of dimension n. We show that these two intrinsic topological properties remain invariant under a measurable homeomorphism and measure invariant map.

is called as measure chart, denoted by ((U, 1 U), ) satisfying following conditions:

1. is homeomorphism,

2. is measurable,

3. is measure invariant

   1 U,

   1 U

   Keywords: Convergence, connectedness, complete measure manifold, local path connected – a.e., interconnected – a.e., maximal connected – a.e.

   1. INTRODUCTION

      In this paper to investigate the intrinsic properties of

      then, the structure ((U, 1 U , 1 U, 1 U),) is called as a measure chart.

      Definition 2.3: Measurable Atlas [7][8][9]

      By an n measurable atlas of class Ck on M we mean a

      countable collection (, 1 , 1 ) of n-dimensional

      complete measure manifold, we have introduced the role of convergent measurable functions on measure manifold.

      measurable charts (( Ui,

      1 ,

      U

      i

      1 ), i) for all i I on

      U

      i

      Any point p U(M,,,) will be a limit point of the measure manifold. The point p U of (M,,,) carries additional information in terms of the ordered pair ({fn}, f) which induces a measurable set An such that the measure of such set is always positive on measure manifold. If the measure of An is zero and any subset B A, also has

      (M, 1, 1) subject to the following conditions: (a1) (( U , 1 , 1 ), ) = M

      i=1 i U U i

      i i

      that is, the countable union of the measurable charts in (, 1 , 1 ) cover (M, 1, 1)

      (a2)For any pair of measurable charts

      measure zero then the measure manifold is complete. On

      (( Ui, 1 , 1 ), ) and (( Uj, 1

      , 1

      ), ) in

      such complete measure manifold, we have introduced different aspects of connectedness, like, locally path

      Ui Ui i

      Uj

      Uj j

      connected, interconnected, maximally connected and

      (, 1 , 1 ), the transition maps -1 and -1 are

      proved some important results that will generate a class of

      i j j i

      connected measure manifolds.

   2. PRELIMINARIES

      Some of the basic definitions referred are as follows Definition 2.1: Measurable chart [7][8][9]

      1. differentiable maps of class Ck (K 1) that is, i -1 : (U U ) (U U ) (n, , ) j -1 : (U U ) (U U ) (n, , )

         j

         j

         i

         j

         i

         i

         j

         i

         i

         i

         j

         j

         i

         j

         are differentiable maps of class Ck (K 1)

         j

         i

      2. Measurable that is, these two transition maps i -1

         Let (U, 1 , 1 ) (M, , ) be a non-empty

         and j -1 are measurable functions if,

         U U 1 1

         (i) any Borel subset K (U U ) is measurable in (n,

         measurable subspace of (M, 1, 1). If there exists a map,

         i i

         , ), then ( -1)-1 (K)

         j

         U ) is also measurable,

         : (U, 1 , 1 ) (n, , ), satisfying the following

         i j j (Ui j

         n

         U U

         conditions,

         1. is homeomorphism,

         2. is measurable,

      then the structure ((U, 1 U , 1 U), ) is called as a

      measurable chart.

      (ii) any Borel subset S j (UiUj) is measurable in ( ,

      i i i j

      , ), then, (j -1)-1 (S) (U U ) is also measurable.

      Definition 2.4: Measure Atlas [7][8][9][10]

      By an n measure atlas of class Ck on M, we mean a

      countable collection (, 1 , 1 ,1 ) of n-

      Definition 2.2: Measure Chart [7][8][9]

      dimensional measure charts (Ui, 1 U , 1 U

      , 1 ), i)

      U

      i

      i

      i=1

      i

      i

      i

      for all i I on (M, 1, 1, 1) satisfying the following conditions:

      (a1)

      (( Ui, 1 U , 1 U

      , 1 ), i) = M

      U

      i

      that is, the countable union of the measure charts in (,

      1 , 1 , 1 ) cover (M, 1, 1, 1).

      A topological measure space (M, 1, 1, 1) that is measurable homeomorphism and measure invariant to a

      measure space (n, , , ) endowed with two structural

      (a2)

      for

      any

      U

      pair of measure charts

      relations between any two atlases 1 and 2 Ak(M):

      U

      (( Ui,

      1 ,

      i

      1 ,

      i

      1 ), i) and

      U

      i

      i) 12, if 12 Ak(M)

      j

      j

      j

      (( Uj, 1 U , 1 U

      , 1 U ), j) in (,1 , 1 , 1 )

      ii) 12 if 1(1) = 1(2)

      then, (M, 1, 1, 1) is called as a measure manifold.

      j

      j

      i

      the transition maps i -1 and -1 are Definition 2.7: Complete measure space (n, , , )

      1. differentiable maps of class Ck (k 1) that is,

         j

         j

         i

         j

         i

         i

         j

         i -1 : (U U ) (U U ) (n, , , ) ,

         [4][5][12]

         Let (n, , , ) be a measure space of dimension n.

         i

         i

         i

         j

         j

         i

         j

         j -1 : (U U ) (U U ) (n, , , ) are Suppose that for every Borel subset U (n, , , ), (U)

         differentiable maps of class Ck (k 1),

      2. measurable and measure invariant. That is,

         j

         j

         1. any Borel subset K i (UiUj) is measurable and measure invariant in (n, , , ) then (i -1)-1 (K) j(UiUj) is also measurable and measure invariant, that is, ((i -1)-1 (K)) = (K).

         2. for any Borel subset S j(UiUj) is measurable and measure invariant in (n, , , )

      then ( -1)-1(S) (U U ) is measurable and

      = 0 and every V U, (V)=0 then (n, , , ) is a complete measure space.

   3. CONVERGENCE ON MEASURE SPACE (n,,, ) Let us consider a measure space (n, , , ) and now we shall discuss some modes of convergence that arise from measure theory.

      Let fn, f : (n, , , ) , be measurable functions on measure space (n, , , ).

      j i i i j

      i

      measure invariant, that is (K) = (j -1)-1 (S).

      (a ) For any two measure atlases ( ,1 , 1 , 1 )

      The following modes of convergence on measure space are discussed in [1], [4], [5], [12]:

      3

      and

      1 1

      1

      1

      1. We say that fn converges to f point wise if, for every x

         X, fn(x) converges to f(x). In other words, for every > 0

         2

         2

         2

         (2, 1 , 1 , 1 ), T : 1 2 is measurable and

         and x X, there exists N (that depends both and x) such that |fn(x) f(x)| whenever n N.

         measure invariant, that is,

         1

         1. if E is measurable in 2 then T

            -1(E) is measurabl in

      2. We say that fn converges to f uniformly, for every > 0, there exists N such that, for every n N, |fn(x) f(x)| ,

      1

      (ii) 1/1 (T

      -1(E)) = /2 (E), if 1 2 .

      for every x X. The difference between uniform convergence and point wise convergence is that with the

      (a4) For any atlas , T, T-1: are invertible measure

      preserving transformations.

      An n, measure atlas is said to be of class C if it is of class Ck for every integer k.

      Let Ak(M) denotes the set of all n measure atlases of class Ck on (M, 1,1,1).

      Definition 2.5: Equivalence Relations in Ak(M)[7]

      1. Two measure atlases 1 and 2 in Ak(M) are said to be

         former, the time N at which fn(x) must be permanently close to f(x) is not permitted to depend on x, but must instead be chosen uniformly in x.

         Now, we discuss some of the modes of convergence on measure space (n, , , ):

         1. We say that fn converges to f point wise almost everywhere if, for (-a.e.) almost everywhere x (n, ,,

            ), fn(x) converges to f(x).

         2. We say that f converges to f point wise almost

            equivalent if in Ak(M). In order that be a n

            1 2 1 2

            1

            V U U

            member of Ak(M) we require that for every measure chart ((Ui, 1 , 1 , ), i) 1 and for every measure

            i i i

            chart

            uniformly if, for every > 0, there exists an exceptional set E for measure (E) such that, fn converges uniformly to f on the complement of E.

         3. We say that fn converges to f in measure if, for every

         > 0, the measures ({x (n, , , ) : |fn(x) f(x)| })

         V

         V

         V

         (Vj,

         1 ,

         j

         1 ,

         j

         1 ), j) 2 the set of i(Ui Vj)

         j

         converge to zero as n .

         According to Christos Papachristodoulous [1], each pair

         and j (UiVj) be open and measurable in (n,, , ) and maps -1 and -1 be of class Ck and are

         ({fn}, f) induces a double sequence of measurable sets

         An ({fn},f) or simply An , n N where An = { x (n, , ,

         i j j i

         measurable. The relation introduced is an equivalence

         relation in Ak(M) and hence partitions Ak(M) into disjoint equivalence classes. Each of these equivalence classes

         ) : |fn(x) f(x)| }, determining the behavior of the pair

         ({fn}, f) with respect to convergence. More precisely, some results are as follows:

         induces a differentiable structure of class Ck on (M, , ,

         (1) f

         f for each j N, lim

         ( A

         ) = 0.

         j=1

         1 1 n

         n n

         (

         1). Any two atlases are compatible, that is, 12 if ( ) in Ak(M).

         (2) {{ x (n, , , ) : fn(x) f(x) } =

         n=1

         En ),

         1 2 where

         k=n

      2. Also any two atlases are compatible, that is, 12 if

      ( ) = ( ).

      En =

      An = {x (n, , , ) : k n : |fk(x) f(x)|

      1 1 1 2

      Definition 2.6:

      }

      (3) f

      ae

      n f for each j N, limn ( n=1 En ) = 0.

      If {fn} is a sequence of measurable functions on (n, , (ii) (S) = 0, if | (fn )(p) – (f )(p) | , n

      1

      , ) converging to f point wise almost everywhere on (n, N,

      , , ), then the ordered pair ({fn}, f) induces a Borel subset An satisfying the following conditions:

      An ={x (n, , , ) : | fn(x) – f(x) | < } n N , where,

      (i) ( An ) > 0, if | fn(x) – f(x) | < , n N,

      (ii) ( An ) = 0, if | fn(x) – f(x) | , n N, that is, ( An) = 0 as n.

      Definition 3.1: Dark point of (n, , , )

      that is, 1(S) = 0 as n .

      Note: Now onwards we denote the Borel subset 1(An )

      of (U, ) by S.

      Definition 3.4:

      The point (f )(p) S (U, ) is called as a dark point in the chart (U, ) if

      1

      (f )(p) (f )(p) in S and (S) = 0.

      The point f(x) An

      n

      is called as a dark point, if fn(x) f(x)

      in An and ( An) = 0.

      Definition 3.5: Convergence 1-a.e. on (M, 1, 1, 1)

      1a.e n

      If (n, , , ) be a measure space and {fn} be a sequence of measurable functions and we say that f converges to f

      Let fn f in An ( , , , ) and if any measure manifold (M,1,1,1) is measurable homeomorphic to (n,

      n n , , ) then for every A

      (n, , , ) 1(A

      ) = S

      a.e. in measure on ( , , , ), and the ordered pair

      n n

      (U,) (M, , ,

      ({fn},f) induces a Borel subsets An , satisfying the

      1

      a.e

      1 1) such that,

      following conditions:

      For any n, An ={x (n, , , ): | fn(x) – f(x) | < }, n

      N}, such that,

      (i) ( An ) > 0, if | fn(x) – f(x) | < , n N,

      (ii) (An ) = 0, if | fn(x) – f(x) | , n N, that is

      ( An ) = 0 as n.

      Suppose ( An ) = 0, then fn(x) f(x) on (n, , , ) then

      k n N such that fk(x) f(x), k. This implies, the

      (fn ) 1 (f ), x on S (U,) (M, 1, 1, 1) and

      S = { p (M, 1, 1, 1) : | (fn )(p) – (f )(p) | < , n

      N}with

      (i) 1(S) > 0, if | (fn )(p) – (f )(p) | < , n

      N,

      (ii) 1(S) = 0, if | (fn )(p) – (f )(p) | , n

      N,that is, 1(S) = 0.

      k=n

      measure of En = Ak is zero, k n, and the region En

      in (n, , , ) is designated as a dark region. To study this dark region where (En ) = 0, we consider the intersection of En whose measure can also be zero.

      Definition 3.6: Complete Measure Manifold

      If (M,1,1,1) is a measure manifold of dimension n and suppose that for every measure chart (U,) (M,1,1,1),

      That is, we undertake the study of (En ) and ( En ).

      =

      A

      If En

      k=n k

      n=1

      = { x (n, , , ) : k n N : |

      1(U)=0 and every V (U,), if 1(V) = 0, then (M,1,1,1) is called as a complete measure manifold.

      n=1

      fk(x) – f(x) | } with (En ) = 0 and En =

      (

      n=1

      n=1

      k=n

      Ak ) , (

      En ) = 0.

   4. DIFFERENT ASPECTS OF CONNECTEDNESS 1-

a.e. PROPERTY ON COMPLETE MEASURE

n=1

Definition 3.2: The Borel set

, , ) if,

En is called as the dark region of (n,

MANIFOLD

In this section, we study the concept of connectedness 1-

a.e. like locally path connectedness 1-a.e., interconnected

(

n=1

En ) = 0.

1-a.e. and maximally path connected 1-a.e. on complete

Now, we shall extend the study of above modes of

convergence of (n,,,) to a measure manifold (M, 1, 1, 1) which is measurable homeomorphic to (n,,,)

measure manifold introduced by S. C. P. Halakatti ([7],[8],[9],[10],[11]).

Let (M, , , ) be a complete measure manifold of

that is (n, , , ) is a complete measure space.

Definition 3.3: Convergence point wise almost everywhere on (M,1,1,1)

Let fn f point wise almost everywhere in (n, , , )

1 1 1

dimension n which is measurable homeomorphic to a measure space (n, , , ). Let {fn}, {gn} be measurable real valued functions converging to f and g respectively in (n, ,,).

Since is measurable homeomorphism from (M, , ,

1 1

and if any measure manifold (M, 1, 1, 1) is measurable homeomorphic to (n, , , ) then for every x An

(n, , , ) 1 x = p 1 (An ) denoted by,

1) to (n, , , ) for every {fn} and {gn} on (n, , , ) there exist corresponding measurable real valued functions

{f } and {g } converging to f and g on (M, ,

S = 1(A ) (U, ) (M, , , ) such that (f

n 1

o )

n 1 1 1 n

f point wise a.e. in S (U, ) (M, 1, 1, 1) such

that

S =1 (An ) = {1 x = p(M,1,1,1): | (fn )(p) – (f

o )(p)|< , n N} on the chart (U, ) satisfying the following conditions:

(i) 1(S) > 0, if | (fn )(p) – (f )(p) | < , n

N,

1, 1).

The ordered pair ({fn }, f ) induces a Borel subset S (U, ) (M, 1, 1, 1) satisfying the following condition:

S = {p (M, 1, 1, 1): | (fn ) (p) – (f ) (p) | < , n

N} on the chart (U, ) for which 1(S) > 0.

Definition 4.1: Locally path connected 1 a.e. on complete measure manifold

The Borel subset S is locally path connected 1 a.e. if a

C – map : [0, 1] S (U, ) such that

(0) = p S ,

(1) = q S, such that 1(S) > 0.

That is, p is locally path connected 1 a.e. to q in S (U, ) Ak(M).

That is, locally path connectedness 1 a.e. is between two points in the same chart (U,) Ak(M).

If 1(S) = 0, then there does not exist a path between p and q.

Definition 4.2:

If 1(S) = 0 where (fn ) f , then S (U, ) (M,

1, 1, 1) is called as a dark region in the chart (U, ).

Let (M, 1,1,1) be a complete measure manifold on which {fn } and {gn } are sequence of real valued measurable functions converging to f and g pointwise 1 a.e. on (U, ) and (V, ) belonging to the atlas respectively. The ordered pairs ({fn }, f ) and ({gn }, g ) induce two Borel subsets S (U, ) Ak(M) and R (V, ) Ak(M) satisfying the following condition:

S = {p (M, 1, 1, 1): | (fn ) (p) – (f ) (p) | <

, n N} on the chart (U, ) for which 1(S) > 0.

R = {q (M, 1, 1, 1): | (gn ) (q) – (g ) (q) | <

, n N} on the chart (V, ) for which 1(R) > 0.

Note: We denote the Borel subsets 1(An) = S (U,)

(M,1,1,1) and 1 (Bn ) = R (V, ) (M, 1, 1, 1).

Definition 4.3: Interconnected 1 a.e on complete measure manifold

The Borel subset S (U, ) (M, 1, 1, 1) is interconnected to the Borel subset R (V, ) (M,

1, 1, 1) 1 a.e. if a C – map

: [0, 1] S R Ak(M) such that

(0) = p S ,

(1) = q S , such that 1(S) > 0 and 1(R) > 0.

That is, p is interconnected 1 a.e. to q in S R

Ak(M).

That is, interconnectedness 1 a.e. is between two charts in the same atlas Ak(M).

If 1(S) =0 and 1(R) =0 then a path between p and q.

Definition 4.4:

If 1(S)=0 where {fn } f in (U,) and 1(R)=0 where {gn } g in (V,), then S is called as dark region in the chart (U, ) and R is called as dark region in the chart (V, ) belonging to the same atlas in Ak(M).

An = { x (n, , , ) : | fn(x) – f(x) | < } , n N, where

(i) (An ) > 0, if |fn(x) – f(x) | < , n N,

(ii) (An ) = 0, if |fn(x) – f(x) | , n N, that is, (An ) = 0 as n.

Similarly,

(i)Bn = { y (n, , , ) : | gn(y) – g(y) | < }, n N, where

(Bn ) > 0, if | gn(y) – g(y) | < , n N,

(ii) (Bn ) = 0, if | gn(y) – g(y) | , n N, that is, (Bn ) = 0 as n and

Cn = { z (n, , , ) : | hn(z) – h(z) | < }, n N, where

(i) (Cn ) > 0, if | hn(z) – h(z) | < , n N,

(ii) (Cn ) = 0, if | hn(z) – h(z) | , n N, that is, (Cn ) = 0 as n.

If (M, 1, 1, 1) is a complete measure manifold that is measurable homeomorphic and measure invariant to (n,

, , ). Then a measurable homeomorphism and measure invariant transformation

: (M, 1, 1, 1) (n, , , ), such that, {fn }, {gn } and {hn } are sequences of real valued measurable functions converging to f , g and h point wise 1-

a.e. on (U,), (V, ) and (W, ) belonging to the atlases i,

j, l respectively. Also, for every induced Borel subsets An , Bn and Cn in (n, , , ), the corresponding induced Borel subsets namely

S (U, ) i Ak(M), R (V, ) j Ak(M) and

Q (W, ) l Ak(M) on (M, 1, 1, 1). Now, we define S, R and Q as follows:

S = {p (M, 1, 1, 1): | (fn ) (p) – (f )(p) | < , n

N} on the chart (U, ) i Ak(M), for which 1(S) >

0,

R = {q (M, 1, 1, 1): | (gn ) (q) – (g ) (q) | <

, n N} on the chart (V, ) j Ak(M), for which

1(R) > 0 and

Q = {r (M, 1, 1, 1): | (hn ) (r) – (h )(r) | < , n

N} on the chart (W, ) l Ak(M), for which 1(Q) >

0.

Note: We denote the Borel subsets 1(An ) = S(U, ) i

Ak(M), 1(Bn )= R(V, ) jAk(M) and

1 (Cn )=Q(W, ) l Ak(M).

Definition 4.5: Maximal connected 1-a.e on complete measure manifold

Let (M, 1, 1, 1) be a complete measure manifold and let

i, j and l Ak(M) be atlases on (M, 1, 1, 1). Let S, R and Q be Borel subsets of i, j and l. Then, we say that Ak(M) (M, 1, 1, 1) is maximally connected if a map

:[0,1] S R Q i j l Ak(M) such that, (0) = p S (U, ) i Ak(M) for which 1(S) > 0, (1) = q R (V, ) j Ak(M) for which 1(R) > 0 and

Let (n, , , ) be a measure space and {f }, {g }and 2 k

n n (1) = r Q (W, ) l A (M) for which 1(Q) > 0.

{hn}are sequences of measurable functions on (n, , , ) converging to f, g and h point wise almost everywhere on

(n, , , ). The ordered pairs ({fn},f), ({gn}, g) and

That is, for each p (U, ) i is path connected to each q

(V, ) j, for i j Ak(M), 1(i j) > 0 for each q (V, ) is path connected to each r (W, )

j l

({hn}, h) induce the following Borel subsets An , Bn and Cn .

We define Borel subsets

Ak(M) and for j l Ak(M), 1(j l) > 0. Then, if

for each p (U, ) i is path connected to each r (W,

) l Ak(M) and for i l Ak(M), 1(i l) > 0 then ( i j l) Ak(M) (M, 1, 1, 1) is maximally path connected if 1(i j l) > 0 on complete measure manifold.

If 1(S) = 0 and 1(R) = 0 then there does not exist a path between p S and q R.

Definition 4.6:

If 1(S) = 0 where {fn } f in (U, ) and 1(R)=0 where {gn } g in (V, ) and 1 (Q) = 0 where {hn } h in (W, ), then S is called as dark region in the chart (U, ) i, R is called as dark region in the chart (V, ) j and Q is called as dark region in the chart (W, )

l in Ak(M).

Now, we show that locally path connectedness is invariant with respect to measurable homeomorphism and measure invariant on a complete measure manifold if 1(S) > 0.

Theorem 4.7: Let (M1, 1, 1, 1) and (M2, 2, 2, 2) be complete measure manifolds of dimension n and m respectively. If M1 is locally path connected 1-a.e in S M1, 1(S) > 0, then a measurable homeomorphism and measure invariant map F: M1 M2 such that M2 is also

locally path connected 2-a.e in F(S) M2 with 2(F(S)) > 0.

Proof: Let (M1, 1, 1, 1) be a complete measure manifold and it is locally path connected. Let (U1, 1) be a measure chart in (M1, 1, 1, 1) that is, S M1 and 1(S) > 0.

Consider a measurable homeomorphism and measure invariant map

F: (M1, 1, 1, 1) (M2, 2, 2, 2) defined for M1, F() M2.

To prove that (M2, 2, 2, 2) is locally path connected, let us consider

= {p (M1, 1, 1, 1): | (fn ) (p) – (f ) (p) | <

, n N} on the chart (U, ) for which 1() > 0.

Now, for every (M1,1,1,1), F() (M2,2,2,2): | F(fn ) (p) – F(f )(p)|<, n N} for which 2(())

> 0.

Let (U, ), (V, ) be charts in M1 and M2 respetively and F(p1), F(p2) F() (M2,2,2,2).

Since F is measurable homeomorphism and measure invariant map, there exists 1 such that for every measure chart (V, ) (M2, 2, 2, 2) such that 2(()) > 0.

M1 M2

(U, ) F

. p1

. p2

(V,)

. F(p1)=q1

. F(p2)=q2

0 1

F

Fig. 1

That is, for every F(p1), F(p2) (V, ) (M2, 2, 2, 2) , there exists,

F1 (F(p1)) = p1 (U, ) (M1, 1, 1, 1).

F1 (F(p2)) = p2 (U, ) (M1, 1, 1, 1).

But (M1, 1, 1, 1) is locally path connected

Therefore, a C map : [0,1] (M1, 1, 1, 1) such that

(0) = p1 S (U, ), 1(S) > 0

(1) = p2 S (U, ), 1(S) > 0.

If 1(S) = 0 then p1 is not locally path connected to p2. Now, since F is homeomorphism a map F : [0, 1] (M2, 2, 2, 2)

such that, F(0) = F(p1) = q1 (V, ), 2(F()) > 0.

F(1) = F(p2) = q2 (V, ), 2(F()) > 0.

If 2 (F()) = 0 then q1 is not maximally path connected to q2.

Therefore, q1 is locally path connected 1-a.e to q2 by F in F() (V, ) (M2, 2, 2, 2). If q1 is locally path

connected 1-a.e. to q2 by F in F() (V,)

(M2,2,2,2) then (M2,2,2,2) is locally path connected.

Therefore, if (M1, 1, 1, 1) is locally path connected 1-

a.e. then (M2, 2, 2, 2) is also locally path connected 2- a.e.

Hence, local path connectedness is invariant with respect to measurable homeomorphism and measure invariant map, if

1() > 0, 2(F()) > 0.

Now, we show that inter connectedness is invariant with respect to measurable homeomorphism and measure invariant on a complete measure manifold if 1() > 0 and 1() > 0.

Theorem 4.8:

Let (M1, 1, 1, 1) and (M2, 2, 2, 2) be complete measure manifolds of dimension n and m respectively. If (M1,1,1,1) is interconnected 1-a.e. in

Ak(M) with 1( )>0 then a measurable homeomorphism and measure invariant map F:(M1,1,1,1) (M2,2,2,2) such that M2 is also interconnected

1-a.e. in ( ) Ak(M) with 2 F( ) > 0. Proof: Let (M1, 1, 1, 1) and (M2, 2, 2, 2) be complete measure manifolds of dimension n and m respectively.

Suppose (M1, 1, 1, 1) is interconnected 1-a.e. in

(M1, 1, 1, 1) with 1() > 0, 1() > 0.

Consider a measurable homeomorphism and measure invariant map

F: (M1, 1, 1, 1) (M2, 2, 2, 2) defined as follows: for any two Borel subsets

= {p1 (M1, 1, 1, 1): | (fn ) (p1) – (f ) (p1) |

<, n N} on the chart (U1, 1) for which 1() > 0

and = {p2 (M1, 1, 1, 1): | (fn ) (p2) – (f ) (p2) |

< , n N} on the chart (U2, 2) for which 1() > 0.

If 1() =0 and 1() = 0 then p1 is not interconnected to p2. There exist

F() = {q1 (M2, 2, 2, 2) : | F(fn ) (q1) – F(f ) (q1) |

< , n N} on the chart (V1, 1) for which 2(()) >

0 and

F() = {q2 (M2, 2, 2, 2) : | F(fn ) (q2) – F(f ) (q2)

| < , n N} on the chart (V2, 2) for which

2(()) > 0.

As (M1, 1, 1, 1) is interconnected 1-a.e then a C – map : [0,1] such that, for every , (M1,1, 1,1)

F(),F() as ascribed in (M2,2,2, 2) such that F : [ 0,1 ] (M2, 2, 2, 2) such that

F 0 = q1 (V1, 1) (M2, 2, 2, 2) , 2(())

> 0,

F 1 = q2 (V2, 2) (M2, 2, 2, 2) , 2(())

> 0.

This implies q1 is interconnected to q2 in (M2, 2, 2, 2).

M1 M2

F

(U1,1)

.p1

(U2,2)

.q1

(V1,1)

(V2,2)

1 .p2 .q2

Fo

0 1

Fig 2

If 2(()) = 0 and 2(()) = 0 then q1 is not interconnected to q2.

Therefore, if (M1, 1, 1, 1) is interconnected 1-a.e. then (M2, 2,2, 2) is interconnected 1-a.e.

Hence, interconnectedness is invariant with respect to measurable homeomorphism and measure invariant transformation if

1(S) > 0, 1(R) > 0 and 2(F(S)) > 0, 2(F(R)) > 0.

Now, we show that maximal path connectedness is invariant with respect to measurable homeomorphism and measure invariant on complete measure manifold if 1(S) > 0 and 1(R) > 0.

Theorem 4.9:

Let (M1, 1, 1, 1) and (M2, 2, 2, 2) be complete measure manifolds of dim n and m respectively. If (M1, 1, 1, 1) is maximally path connected 1 – a.e. in i j

l of Ak(M1) such that 1(S)> 0, 1(R)>0 and 1(Q)>0 and

Proof: Let (M1, 1, 1, 1) and (M2, 2, 2, 2) be complete measure manifolds of dimension n and m respectively.

Suppose (M1, 1, 1, 1) is maximally path connected 1- a.e. in i j l Ak(M1) with i, j, l Ak(M1): 1(S)

> 0, 1(R) > 0 and 1(Q)> 0.

Consider a measurable homeomorphism F:(M1,1,1,1)(M2,2,2,2). Since (M1, 1,1,1) is maximally path connected 1-a.e. in i j l Ak(M), then three Borel subsets S, R and Q such that,

S = {p1 (M1, 1, 1, 1): | (fn ) (p1) – (f )(p1) | <

, n N} on the chart (U1, 1) i Ak(M), for which

1(S) > 0 and

R = {p2 (M1, 1, 1, 1): | (gn ) (p2) – (g )(p2) | <

, n N} on the chart (U2, 2) j Ak(M), for which

1(R) > 0 and

Q = {p3 (M1, 1, 1, 1): | (hn ) (p3) – (h )(p3) | <

, n N} on the chart (U3, 3) l Ak(M), for which

1(Q) > 0.

Then, there exist a path :[0,1] S R Q

if a measurable homeomorphism and measure invariant

Ak(M)

i j l

map F: M1M2 then M2 is also maximally path connected

2-a.e. in i j l of Ak(M2) such that 2(F(S)) > 0,

2(F(R)) > 0 and 2(F(Q)) > 0

in a complete measure manifold (M1, 1, 1, 1) such that (0) = p1 S (U1, 1) i Ak(M) for which 1(S) > 0,

2

(1) = p2 R (U2, 2) j Ak(M) for which 1(R) > 0 and

(1) = p3 Q (U3, 3) l Ak(M) for which 1(Q) > 0.

where each p1 S (U1, 1) i is maximally path connected to each p R (U , ) and each p R

p1 S (U1, 1) i (M1, 1, 1, 1) there exist q1

F(S) (V1, 1) i (M2, 2, 2, 2) such that,

F S ={ F(p1) = q1 (M2, 2,2, 2): | F(fn ) (p1) – F(f

)(p1) | <,

n N} on the chart (V1, 1) i Ak(M), for which

2 2 2 j 2

(U2, 2) j is maximally path connected to each p3 Q

(U3, 3) l in i j l Ak(M).

If 1(S) = 0, 1(R) = 0 and 1(Q) = 0 then p1 is not maximally path connected to p2 and p2 is not maximally path connected to p3.

Since F is measurable homeomorphism and measure invariant, for

2(F(S)) > 0.

Similarly, for every p2 R (U2, 2) 2 in

(M1, 1, 1, 1) there exist q2 F(R) (V2, 2) j

(M2, 2, 2, 2) such that,

F(R)={ F(p2) = q2 (M2, 2, 2, 2): | F(gn ) (p2) – F(g

)(p2) | <,

n N} on the chart (V2, 2) j Ak(M), for which

2(F R ) > 0

(U1, 1)

.p1

i

M1 M2

F

i

(V1, 1)

.q1

(U2, 2) (V2, 2)

. p2

j

j

. q2

(U3, 3)

.p3

l

F

0 1

k

(V3, 3)

.q3

and similarly, for every p3 (U3, 3) j in (M1, 1, 1, 1) there exist q3 () (V3, 3) l (M2, 2, 2, 2) such that,

()={ (p3)=q3(M2, 2, 2, 2): |(hn )(p3) – (h )(p3) |<, n N} on the chart (V3, 3) l Ak(M), for which 2( ) >0.

Since F is measurable homeomorphism and measure invariant, for every :[0,1] i j l Ak(M) in (M1, 1, 1, 1) which connects p1, p2 and p3 maximally, there exist a corresponding path F : [0, 1]

F() F() F() in(M2, 2, 2, 2) which connects maximally F(p1) = q1 to F(p2) = q2 and F(p2) = q2 to F(p3) = q3 in

Bi Bj Bl Ak(M2) satisfying 2(()) > 0,

2(()) > 0 and 2(()) > 0 then q1 is maximally path connected to q2 and q2 is maximally path connected to q3.

Fig. 3

Therefore, we have shown that if (M1, 1, 1, 1) is maximally path connected then (M2, 2, 2, 2) is also maximally path connected.

Hence, maximal path connectedness 1-a.e. is invariant with respect to measurable homeomorphism and measure invariant function F.

The above theorem shows that if 1() = 0, 1() =0 and

1() =0 are dark regions in the respective charts (U1,1), (U2,2) and (U3,3) in (M1,1,1,1) then 2(()) = 0, 2(()) = 0 and 2(()) = 0 are dark regions in the corresponding charts (V1,1), (V2, 2) and (V3, 3) in (M2,

2, 2, 2).

5 CONCLUSION

In this paper S.C.P. Halakatti has investigated two intrinsic properties on complete measure manifold. We have shown that locally path connected 1- a.e. property, interconnected 1- a.e. property and maximally path connected 1- a.e. on complete measure manifold are invariant under measurable homeomorphism and measure invariant map.

The above study on different aspects of connectedness on complete measure manifold vindicates that, the local path connectedness, the inter connectedness and the maximally path connectedness are invariant under measurable homeomorphism and measure invariant function F.

One can show that if F: (M1,1,1,1) (M2,2,2,2), G:(M2,2, 2, 2) (M3,3,3,3), then the composite function G F: (M1,1,1, 1) (M3,3,3,3), satisfies equivalence relation on a complete measure manifold paving a way for a new manifold called network manifold. We carry the study on such network manifolds in our future work.

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