International Journal of Advancements in Research & Technology, Volume 3, Issue 9, September ISSN

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1 Iteratoal Joural of dvacemets Research & Techology, Volume 3, Issue 9, September Topologcal Propertes o Measure space(r, T wth Measure codto S. C. P. Halakatt 1 ad H. G. Halol 2 1 Departmet of Mathematcs, aratak Uversty Dharwad, arataka, Ida. 1 Emal:-scphalakatt07@gmal.com 2 Research studet, aratak Uversty Dharwad, bstract: study of some topologcal propertes o measure space (R, T wth measure codtos, has bee troduced to develop the exteded versos of Hee-Borel property (HBP, coutable compactess, Ldelof, fte tersecto property, Bolzao-Weerstrass property. The ter-relatoshp of these exteded propertes are also studed. eywords: e- Hee-Borel property,e- Coutably compactess, e-ldelof, e- Bolzao-Weerstrass property, measure space, σ addtvty, mootoocty ad σ- sub-addtvty property. 1. Itroducto: I ths paper S. C. P. Halakatt has cosdered some propertes of topologcal space ad has troduced measure codtos o these topologcal propertes. The author has also developed the exteded versos of Hee-Borel Property (HBP, Ldelof, fte tersecto property, coutable compactess, Bolzao- Weerstrass property o a measure space (R, T, Σ, adoptg geeral measure codtos lke σ- addtvty, σ- subaddtvty, mootocty. The exteded verso of these propertes are redesgated versos of topologcal propertes o (R, T.Ths study has a terestg mplcatos o the measure mafolds, the cocept whch was troduced by S. C. P. Halakatt [2]. It s terestg to kow that the varace of these propertes uder the measure varat trasformatos from measure mafolds (M, T111 oto (R, T erches the further study of aalyss o the measure mafolds. The applcatos of such measure varat propertes o a measure mafold are the feld of Egeerg scece, Neuroscece ad Bra Scece. I our prevous paper [2] we have troduced some basc defto o measure mafold. Defto 1.1 Measureable Chart [2] Let (U, T 1 U (M, T11 be a o empty measurable subspace of (M, T11 f there exsts a map, φ: (U, T 1 U φ (U, T 1 U (R, T, satsfyg the followg codtos, ( φ f homeomorphsm, ( φ s measurable f for every measurable Vϵ (R, T, φ -1 (V ϵ (M, T11. The ((U, T 1 U, φ s called a measurable chart. Defto 1.2 Measure Chart [2] measurable chart ((U, T 1 U, φ equpped wth a measure μ 1 U s called a measure chart, deoted by ((U, T 1 Σ 1 U U U, φ satsfyg followg codto, ( φ f homeomorphsm, ( φ s measurable fucto f for every measurable Vϵ (R, T,, φ -1 (V ϵ (M, T111 s measurable, ( φ s measure varat. The, the structure ((U, T 1 U U,φ s called a measure chart. Defto 1.3 :- Measurable tlas [2] By a R measurable atlas of class C k o M we mea a coutable collecto (, T 1 of - dmesoal measurable charts ( (, T 1, Σ 1, φ for all ϵ N o (M, T11 subject to the followg codtos: (a1 =1 U, T 1, ϕ = M.e. the coutable uo of the measurable charts (, T 1 cover (M, T11

2 R = Vj Iteratoal Joural of dvacemets Research & Techology, Volume 3, Issue 9, September (a2 For ay par of measurable charts U, T 1, ϕ ad (a2 For ay par of measure charts U, T 1, ϕ ad Uj, T 1 Uj 1 Uj, ϕ (, T 1, the Uj, T 1 Uj 1 Uj 1 Uj, ϕj ( trasto maps φr o ϕj -1 ad φrj o φ -1 are, T 1, the trasto maps φr o ϕj -1 ad φrj o ( dfferetable maps of class C k ( 1 φ -1 are.e. φ o φrj -1 : φrj ( Uj φr (U Uj ( dfferetable maps of class C k ( 1 (R, T.e. φ o φrj -1 : φrj ( Uj φr (U Uj φj o φr -1 : φr (U Uj φrj ( Uj (R, T,Σ, (R, T φj o φr -1 : φr (U Uj φrj ( Uj are dfferetable maps of class C k ( 1 (R, T,Σ ( measurable, are dfferetable maps of class C k ( 1.e. these two trasto maps φr o ϕj -1 ad φrj o ( measurable. φ -1 are measurable fuctos f,.e. These two trasto maps φr o ϕj -1 ad φrj o (a for ay measurable subset φr (U Uj, φ -1 are measurable fuctos f (φr o φrj -1-1 ( φrj ( Uj s also measurable, (a for ay measurable subset φr (U (b for ay measurable subset S φrj ( Uj, Uj,(φR o φrj -1-1 ( φrj ( Uj s also (φrj o φr -1-1 (S φr (U Uj s also measurable. measurable, (b for ay measurable subset S φrj ( Uj, (φrj o φr -1-1 (S φr (U Uj s also Defto 1.4:- Restrcto of Measure μ1 o (, T 1 measurable,. [2] (a3 For ay two measure atlases ( 1, T 1 1, Let (M, T111 be a measure space ad let (, T 1 Σ 1 1 ϵ (M, T11, 1 be a o-empty 1 1 ad ( 2, T , we say measureable tlas. The measure μ 1 that a mappg,t : 1 2 s measurable f T o (, T 1 (E s measurable for every measure chart 1 s called the restrcto of measure μ1 o (, T 1 E = ((U, T 1 1 U U U, φ ( 2, T 1 2 2,. μ 1 2 ad the mappg s measure preservg f μ 1 1 (T -1 (E = μ 1 R 2 (E,where R1 2 ad μ Defto 1.5:- Measure tlas [2] 1 μ 1 R 1 2. The structure (, T 1 s called The we call T a trasformato. measure tlas f (, T 1 s a measureable (a4 If a measurable trasformato T: tlas equpped wth restrcted measure μ preserves a measure μ 1. 1, the we say that μ 1 s T- varat (or varat uder T. If T s vertble ad f both T ad T -1 are measurable ad measure preservg the we call T a vertble measure preservg trasformato. Codto to be satsfed for Measure tlas:- Defto 1.7:- Measure tlas [2] By a R measure atlas of class C k o M, we mea a coutable collecto (, T 1 of -dmesoal measure charts ( (, T 1, φ for all ϵ N o (M, T11, μ1 satsfyg the followg codtos:- (a1 =1 U, T 1, ϕ = M.e. the coutable uo of the measure charts (, T 1 cover (M, T111. Two measure atlases 1 ad 2 m k (M are sad to be equvalet f (1 R2 m k (M.I order that 1 R2 be a member of m k (M we requre that for every measure chart ((, T 1, φ 1 ad every measure chart (Vj, T 1 Vj 1 Vj Vj, ΨRj 2 the set of φ(u R ad ΨRj ( Vj be ope measurable (R, T ad maps φ o

3 Iteratoal Joural of dvacemets Research & Techology, Volume 3, Issue 9, September Ψj -1 ad φrj o φ -1 be of class C k ad are measurable. The relato troduced s a equvalece relato m k (M ad hece parttos m k (M to dsjot equvalece classes. Each of these equvalece classes s called a dfferetable structure of class C k o (M, T111. measure space (M, T111 together wth a dfferetable structure of class c k s called a dfferetable measure - mafold of class C k or smply a C k measure -mafold. o empty set M equpped wth dfferetable structure, topologcal structure ad algebrac structure σ- algebra s called Measurable Mafold. measure μ1 defed o (M, T11 ad the quadruple (M, T111 s called Measure Mafold. Defto 1.7 Measurable Mafold[2] o-empty set M, whch s modeled o measureable space (R, T s called a measurable mafold deoted by (M, T Prelmares: I ths secto we cosder some basc defto o topologcal space (R, d. Defto 2.1 Local Bass[5][6][1] Let (R,d be a topologcal space ad let p (R,d. local bass at p s a collecto Vp of T such that,for every p ϵ (R,d there exsts B ϵ Vp ad G ϵ T such that: p ϵ B G. Theorem 2.2 [5] famly V of subsets of a set R s a bass for some topology o (R, d f ad oly f: (. (R, d = {B:B ϵ V} ad (. If B 1, B 2 ϵ V ad p ϵ B 1 B 2 the there exsts B ϵ V such that q ϵ B ad B B 1 B 2. Defto 1.8 Measure Mafold [2] If (R, T s a measure space a o-empty set M modelled o (R, T s called Measure Mafold. P B Defto 1.9 Borel Cover [2]. By a Borel cover vz, { =1 : s are Borel sets}, we mea a coutable uo of all Borel sets Fg1 belogg to (R, T. I ths paper The collecto of Borel sets belogg to Σ are deoted by,b,c etc 1.11 Hee-Borel property (HBP o Topologcal Space (R,d For a subset of the topologcal space (R,d, has the Hee- Borel property f every ope coverg of admts a fte sub-coverg. The Hee-Borel property o the measure space (R, T havg two structures oe topologcal ad the other the algebrac structure, σ-algebra- Σ was troduced ad proved our paper [2]. lso the cocept of Borel cover was troduced o measure space (R, T. I ths paper we cosder the exteded verso of exteded Hee-Borel property (EHBP. The advatage of ths exteded verso of Hee- Borel property s that the (R, T aturally admts e-coutable compactess property ad Exteded verso of Bolzao-Weerstrass property (ebwp o (R, T. B 1 B 2 Defto: 2.3 Frst Coutable space[3][5] topologcal space (R, d s frst coutable provded there s coutable local bass at each pot of R. Defto: 2.4 Secod Coutable[5] topologcal space (R, d s secod coutable provded there s coutable bass for T. Defto: 2.5 Compactess [3][5][1] subset of a topologcal space (R, d s compact provded every ope cover of has a fte sub-cover. Defto: 2.6 Coutably Compact[4][6] topologcal space (R, d s coutably compact provded every coutable cover of R has a fte sub-cover. Defto: 2.7 Locally Compact at a pot[5] topologcal space (R, d s locally compact at a pot p R, provded there s a ope set G ad a compact subset of (R,d such that p G.

4 Iteratoal Joural of dvacemets Research & Techology, Volume 3, Issue 9, September Fg 2 G.p ( R,d Defto 2.8 Locally Compact Topologcal Space (R, d. topologcal space (R, d s locally compact f t s locally compact at each of ts pots. Defto 2.9 Ldelof space[5][6] topologcal space (R, d s a Ldelof space, f every ope cover has a coutable subcover Defto: 2.10 Hee- Borel property[hbp][5] subset of a topologcal space (R, d has the Hee Borel property, f every ope coverg of admts a fte sub coverg. Defto 2.11:- Fte Itersecto Property (f..p famly of subsets of a set has the Fte tersecto property, f s a fte sub collecto of the {: ϵ }. Defto 2.12 Bolzao-Weerstrass property [5] topologcal space (R,d has the Bolzao- Weerstrass property provded that every fte subset of (R,d has a lmt pot. Defto 2.13 Measure o (R, T [9] Let (R, T be a measure space. fucto μ: Σ [0, ] s called a measure o (R, T or smply a measure o Σ f, (. μ (ø = 0 (. μ ( =1 = =1 μ( for all dsjot coutable collecto { } of measurable sets Σ. The topologcal space ( R,d alog wth topologcal structure T,a algebrac structure σ- algebra Σ, ad a measure µ o (R, T s called a measure space. Note that, the values of a measure are oegatve real umbers belogg to [0, ]. Property ( of a measure s called σ-addtvty ad sometmes a measure s also called σ- addtve measure. Theorem 2.14[9] Let (R, T be a measure space (. (Mootocty If, B Σ ad B the µ( µ(b (. If, B Σ, B ad µ( <,the µ( B = µ(b - µ(. (. (σ- sub addtvty If 1,2,, Σ the μ( =1 μ( =1 Defto 2.15 [2] Let (R, T be a measure space (. µ s called fte f µ (R < (. µ s called σ-fte f there exsts 1,2,, Σ, so that R = =1 ad µ( < for ϵ N. ( set Σ s called of σ-fte measure f there exsts 1, 2,.., Σ so that =1 ad µ( < for all. I Secto 3, measure ope cover meas Borel cover. 3. Exteded verso of some Topologcal Propertes o a Measure Space (R, T, μ I ths secto Halakatt has developed the exteded versos of Hee-Borel property, coutable compactess, Bolzao-Weerstrass property o a measure space (R, T. It s observed that the measure space carres two structures, oe the topologcal structure T ad the other, the algebrac structure σ-algebra, o whch a measure fucto µ has bee troduced. The frst author otced that a few topologcal propertes lke Hee-Borel property, coutable compactess, Bolzao- Fte tersecto property (f..p, Weerstrass property o a usual topologcal space ( R,d, terestgly admts the exteded versos of these topologcal propertes o a ewly structured space.e., a measure space (R, T admtg σaddtvty, σ-subaddtvty, mootocty. Such a exteso of topologcal property o (R, T, μ are re-desgated as exteded Hee-Borel property(ehbp, exteded Bolzao-Weerstrass property(ebwp, exteded fte tersecto property (e.f..p,

5 B2, Iteratoal Joural of dvacemets Research & Techology, Volume 3, Issue 9, September I ths paper S.C.P. Halakatt has troduced ad developed the exteded versos of some basc deftos o measure space (R, T. Note that the followg symbols are used our deftos ad theorems. Note: 3.1 (1 The collecto of ope sets belogg to topologcal space (R,d are deoted by G.H etc (2 The collecto of Borel sets belogg to the famly subspace of (R, T are deoted by,b,c etc (3 The members of cover for measure space (R, T are deoted by,b,...,etc. (4 Every measure ope cover meas Borel cover. Defto 3.2 Local bass for measure space (R, T:- Let (R, T be a measure space ad let q (R, T the a local bass at a pot q ϵ (R, T s a coutable collecto Vq of a Borel sets Σ satsfyg the followg codtos: (. for each q (R, T, there exst Σ, B Vq such that q B. (. for B, µ (B µ (... (Mootocty (R, T B q. q. Fg 4 Proof: ( Sce, (R, T has a cover G.e. (R, T G for each ϵ I. Every ope set G ϵ T geerates a correspodg Borel set B ϵ Σ. Therefore there exsts a ope cover { B: B ϵ V }.e. Let V be a famly of Borel subsets of (R, T such that V s a bass for (R, T. Therefore (R, T = { B: B ϵ V }. ( Suppose B1,B2 V ad q B1 B2 where V s a bass for Σ, there s a sub collecto V of V such that, B1 B2 = {B: B V } satsfyg σaddtvty (B1 B2 = {μ (B: B V }. Therefore there exsts B ϵ V such that q ϵ B ad B B1 B2, satsfyg the σ subaddtvty property (B μ (B1 R for q ϵ B ad B B1 B2. Defto 3.4 Frst Coutable Measure Space (R, T(e-Frst Coutable: measure space (R, T s frst coutable f there s a collecto Vq of Borel sets as local base at each pot q ϵ (R, T satsfyg the followg codtos: (. for each q (R, T, there exst Σ, B Vq such that q B. (. for B the µ (B µ (...(Mootocty frst coutable measure space s redesgated as e- frst coutable. Fg3 Theorem 3.3 famly V of Borel sets of a measure space (R, T s a bass for (R, T f t satsfes the followg codtos: (. (R, T = {B: B V} ad (. If B1,B2 V ad q B1 B2 the, there exsts B Vq such that q B ad B B1 B2., ad (. µ (B µ (B1 B2, (R, T Defto 3.5 Secod Coutable Measure Space (R, T(e- Secod Coutable. measure space (R, T s secod coutable provded there s a coutable base for all q (R, T satsfyg the followg codtos: (.for each q ε (R, T, µ there exst B Vq ad Σ such that q B for =1,2,3,. (. for B, µ(b µ( secod coutable measure space s redesgated as e- secod coutable B1 B B2 Defto 3.6 collecto ε of a arbtrary subset of a oempty topologcal place R s sad to geerate σ-

6 Iteratoal Joural of dvacemets Research & Techology, Volume 3, Issue 9, September algebra Σ(ε, f the tersecto of all σ-algebra of subsets of R clude ε, amely Σ(ε = { Σ : Σ s a σ-algebra of subsets of R ad ε Σ }, the smallest σ-algebra. Note that, there s at least oe σ-algebra of subsets of R, whch cludes ε ad ths s P(R Defto 3.7 Borel σ-algebra Let R be a topologcal space ad T-the collecto of all ope subsets of R be a topology o R. The σ-algebra Σ geerated by T cotag all ope subsets of R, s called the Borel σ-algebra of R, deoted by BR,.e BR = Σ (J. The elemets of B R are Borel sets R. ( μ (, T =1 = Defto 3.8 Borel Cover [2]. μ ( ( j, T j j j By a Borel cover vz, { =1 : s are Borel sets}, such that, we mea a coutable uo of all Borel sets j I μ ( j, T belogg to (R, T. j j j I ths paper The collecto of Borel sets =1 μ (, T. belogg to Σ are deoted by,b,c coutable compact measure space s redesgated as e- coutable compact. Defto 3.9 Compact Measure Space (R, T(e-Compact: Defto 3.11 Ldelof Measure Space(e- Let (, T (R, T, the Ldelof (, T measure space (R s compact measure, T s a Ldelof measure space f, every Borel cover has a space f every Borel cover of parwse dsjot coutable Borel subcover wth σ subaddtvty sequece of measure sets.e. (, T =1 of (, T property, satsfyg the followg codtos: (. for every cover (, T has a fte Borel sub cover, satsfyg the =1 of a subset (, followg codtos: T of (R, T sce ( for every cover (, T fte sub cover s always coutable sub cover, =1 of a subset (, T of therefore, there exsts a coutable sub cover (R, T the.e. there exsts a fte sub cover whch s also coutable,.e. ( j, T j ( ( j, T j=1 j, j j that covers j j that covers (, T (, T (. μ (, T =1 = ( μ (, T μ ( ( =1 = j, T j j j μ ( ( j, T j j satsfyg σ- subaddtvty property, j satsfyg σ- subaddtvty property, μ ( j I =1 μ (, T j, T j j j. compact measure space s redesgated as e- compact. Defto 3.10 Coutably Compact Measure Space (R, T(e- Coutably Compact measure space (R, T s coutably compact f every coutable Borel cover of (R, T, Σ has a fte Borel sub cover,.e. ( for every cover (, T =1 of a subset (, T of (R, T, the there exsts a fte sub cover,.e. ( j, T j j, that covers j (, T of (R, T, satsfyg σsubaddtvty property, μ ( j I =1 μ (, T j, T j j j. Ldelof measure space s redesgated as e- Ldelof. Defto 3.12 Locally Compact at a pot

7 Iteratoal Joural of dvacemets Research & Techology, Volume 3, Issue 9, September of Measure Space (R, T(e-locally compact measure space (R, T s locally compact measure space at a pot q (R, T f there s a Borel set B Σ ad a compact subset (, T of (R, T such that, (, T (R, T (R, T (. For every q (R, T, there exst B ϵ Σ, T such that q ϵ B (, T, (. μ (B μ ( (Mootoocty.q B.q exteded Hee- Borel property s redesgated as e- Hee- Borel property(ehbp Defto 3.14: Exteded Fte Itersecto Property(eFIP. famly of Borel subset of a measure space (R, T has the exteded fte tersecto property f s a fte sub collecto of the {, where s coutable famly of Borel closed subset of (R, T : ϵ }, satsfyg μ (>0. exteded fte tersecto property s redesgated as e-fp. Defto 3.15 Exteded Bolzao- Weerstrass Property [ebwp]:- measure space (R, T has the Exteded Bolzao-Weerstrass property provded that every fte subspace of measure space (R, T, Σ has a lmt pot, satsfyg the followg codtos: (. for ay fte subspace (, T of (R, T there exst a lmt pot q Fg 5 (, T, (. μ (, T 0, measure space (R, T s locally compact f t s locally compact at each of ts exteded Bolzao-Weerstrass property s pots. redesgated as e- Bolzao-Weerstrass property(ebwp Defto 3.13 Exteded Hee Borel Property (ehbp Let (, T be a measure subspace Theorem:-3.16 of a measure space (R, T, the a measure If Hee-Borel property (HBP holds o subspace (, T has the exteded topologcal space (R, d the, a measure space Hee- Borel property f every ope Borel cover (R, T admts a exteded Hee-Borel of (, T property (ehbp satsfyg admts a fte Borel sub ( If (, cover satsfyg σ-sub addtvty property: T ( If (, T (, T εi the there exsts a (, T εi the there exsts a fte sub cover for j ϵ J, J I, such that (, fte sub cover for j ϵ J, T J I, such that (, T ( j, T ( j, T j j j satsfyg, j j=1 j satsfyg, j (. μ (, T (. μ (, T = μ ( ( j, T j = μ( ( j=1 j j, T j j j j = j=1 μ ( j, T j = j=1 μ ( j, T j j,... j ( σ-addtvty. j j.( σ-addtvty lteratvely, every coutable Borel cover has a fte sub cover satsfyg measure codto.

8 Iteratoal Joural of dvacemets Research & Techology, Volume 3, Issue 9, September Proof:- Suppose (R, dadmts Hee-Borel property the every ope coverg of a subset (R, d admts a fte sub cover. To show that a measure space (R, T admts a exteded Hee-Borel property, t s ecessary to observe that the σ- algebra structure o (R, T allows a coutable Borel ope cover s made up of parwse dsjot sequece of {(, T }for a measure subset (, T of (R, T. Hece to show that (R, T admts a ehbp, t s suffcet to show that every coutable Borel ope cover has a fte Borel sub cover. Let us prove the codtos ( ad ( of the theorem. ( Let (, T, (R, T s a measure sub space of (R, T ad let { (, T =1 cover (, T dsjot sequece {(, T } be a coutable Borel whch are par wse }for a measure subset (, T..e. (, T (, T =1 (1 satsfyg σ-addtvty codto o measure, μ (, T = μ ( (, T =1 = =1 μ(, T (2 Hee-Borel property satsfyg σ-sub addtvty. ( - addtvty Sce R, by Hee-Borel property o (R,d t mples that every ope cover has fte sub cover.vz { j=1 j j=1 j }, such that (3 Sce {: ϵ I} are ope sub sets (R, T ad the ope subsets of T geerate Borel sets therefore the correspodg Borel cover {(, T : ϵ I } s a coutable ope Borel cover for (, T (R, T. Hee-Borel Property o (R, d mples, every ope cover has fte sub-cover, correspodgly, every topologcal measure subspace havg coutable Borel cover has a fte Borel sub cover (, T =1 (, T..(4, satsfyg the σ-subaddtvty property of measure space,vz, μ (, T = ( =1 (, T = =1 μ(, T (5 (σ- addtvty has a fte Borel sub cover, for j ϵ J, J I (, T ( j, T j j..(6 j Hece (1 s satsfed. (2 Equato (6 satsfyg -subaddtvty codto of a measure, vz, μ (, T = ( ( j, T j j : j ϵ J, J I j = j=1 μ ( j, T j j,...(7 j ( σ-addtvty. Ths mples that, every coutable Borel cover has a fte Borel sub cover. Hece exteded Hee-Borel property o (R, T admts coutable compactess o a measure space. Remark 3.17 ( Hee-Borel Property o a topologcal space o (R, T mples that every ope cover has a fte sub cover. If the σ-algebrac structure s duced o a topologcal space (R, T alog wth the measure µ, the σ-algebra structure trasforms a topologcal space (R, T to a topologcal measurable space ad trasforms ope cover of (R, T to a coutable Borel cover of (R, T. If ths coutable Borel ope cover has a fte sub cover the a measure space (R, T admts a exteded verso of ( The cosequece of such exteso of Hee- Borel property o (R, T s that the exteded Hee-Borel property o (R, T admts e-coutable compactess o a measure space. Hece the above theorem ca be restated as fallows. Theorem 3.18 measure space (R, T admttg e-hee- Borel property s e- coutably compact satsfyg σ-addtvty property of a measure μ. Proof: The proof s obvous as Theorem 3.16 Theorem 3.19 measure space (R, T admttg exteded Hee Borel property(ehbp s e- Ldelof satsfyg σ-subaddtvty property. Proof: Suppose (R, T admts ehbp..e every measure Borel coverg made up of parwse dsjot sequece of Borel sets vz, εi ((, T of a measure sub set

9 Iteratoal Joural of dvacemets Research & Techology, Volume 3, Issue 9, September (, T, Σ, μ of (R, T has a fte measure Borel sub-coverg, vz, ( j I j j, T j, j We ow show that, every measure Borel cover has a coutable measure sub cover. ccordg to ehbp every Borel cover has a fte Borel sub cover, but every fte Borel sub cover s always coutable sub cover,satsfyg σ- sub addtvety property. Therefore, for every Borel coverg { εi ((, T } for a subset (, T, Σ, μ of (R, T, there exts a fte Borel sub cover, whch also coutable,.e. { jεj ( ( j, T j j } such that, whe j (, T, Σ, μ the, Σ, ( μ εi, T (, T, j J ( j, T j j j for J I, satsfyg the σ- subaddtve property of measure.e. μ( ( ε1, T = μ( j I ( j, T j j measure codto μ (, j for whch, j I μ ( j T, T j μ { {(, T =0 j j ( =1 μ (, T, T > 0. ( Coversely, Ths proves, every Borel cover has a coutable Borel sub cover satsfyg the σ- subaddtve property of measure. Hece ehbp mples e-ldelof o a measure space Theorem 3.20 measure space (R, T s e-coutably compact f ad oly f every coutable famly of Borel closed subsets of (R, T wth the e- fte tersecto property has a o-empty tersecto,.e. ( f s a fte sub collecto of, where s coutable famly of Borel closed subset of (R, T the { : ϵ }, ( satsfyg the measure codto μ (>0. Proof: Suppose (R, T s e-coutable compact Let = {(, T : ϵ I} be a coutable famly of Borel closed subsets of (R, T wth the exteded fte tersecto property ( Suppose, {(, T =1 = Let = { (R, T - (, T : ϵ I } s collecto of Borel ope subsets of (R, T, Σ. The =1 = Sce {(R, T µ, T } (R, T- {(, T =1 } = (R, T- = (R, T s a Borel cover of (R, T, sce (R, T, Σ s e-coutably compact there exst a fte umbers 1, 2, 3,..., of I such that, j=1 {(R, T - ( j, T j, j μ j : ε I, j = 1,2, } covers (R, T. Thus (R, T = j=1 ({(R, T - ( j, T j j=1 { ( j, T j Ths mples j=1 { j } j = (R, T j } j ( j, T j whch s cotradcto hece ε I j } = j (, T satsfyg μ > 0 } μ (, mples μ Suppose every coutable famly of closed Borel sub sets of (R, T wth the e-fte tersecto property has a o empty tersecto. Let = {(B, T B B B : ϵ I} be a coutable Borel cover of (R, T. Suppose does ot have a fte Borel subcover. Let = {(R, T - (B, T B B B : ϵ I}, The s a coutable famly of Borel closed subsets of (R, T. Let J be a fte subset of I. Sce does ot have a fte Borel sub cover. εj { (R, T - (B, T B B B = (R, T - εj {(B, T B B B } Therefore has the exteded fte tersecto property wth μ (, T > 0. Hece =1 { (R, T - (B, T B B B } Ths s cotradcto because εi {(R, T - (B, T B { B B B } = (R, T - B = εi (B, T B Therefore (R, T s e-coutably compact measure space (R, T s e-coutably compact ff t s e- fte tersecto property

10 Iteratoal Joural of dvacemets Research & Techology, Volume 3, Issue 9, September Theorem 3.21 measure space (R, T admttg e-coutably compact mplctly satsfes the e- Bolzao-Weerstrass property, wth the followg codtos, ( for ay fte subset (, T of (R, T there exst a lmt pot q (, T where (,T 0. lmt pot (R, T. (1 study o exteded versos of some topologcal propertes o a measure space (R, T, Σ lke e-hee-borel property, e-coutable compactess, e-ldelof, e- fte tersecto property ad e- Bolzao- Weerstrass property has show that there exst a relatoshp betwee dfferet forms of exteded versos of compactess o a measure space (R, T that s depcted below, Proof: Let (R, T be a e-coutably compact. Let ehbp e-coutable compactess, T be a fte subset of R ad let, T cotas a coutably fte set of B = {q : N} we may assume that j the q qj. The proof s by cotradcto. Suppose B has o lmt pot ad for each N, e-ldelof e-bwp e-fp (2 Oe ca observe that dfferet topologcal propertes of compactess ad ther terrelatoshp o a measure space (R, T remas varat uder homeomorphsm wth respect to topologcal structure o (R, T. Wth measure codto, I our future work, we wll show that these µ( (C, T C C =1 C > 0.. (2 topologcal propertes whe defed o measure space (R, T, they may acqure measure codtos ad rema varat uder measure But f qk ϵ B, the qk does ot belogs to Ck+1 ad varat trasformato. Ths approach has hece qk does ot belogs to (C, T C C wder applcatos buldg mathematcal =1 C. models for the problem the feld of Therefore, Egeerg scece ad Techology, also the feld of Bra scece ad Neuroscece. C = {q ϵ B: } s a Borel closed set (R, T, μ satsfyg the e-fte tersecto property (e.f..p, (C, T C C =1 C. (1 (C, T C C =1 C = wth satsfes codto of measure s such that µ ( (C, T C C =1 C = µ ( = 0 ad we have a cotradcto to our assumpto. For B=( B, T B B B, =(, T Sce B has a lmt pot q ad B wth σ- addtvty µ (B µ (,.e. q ϵ B such that µ (B µ (, therefore,, T has a Cocluso:- Oe ca otce that some topologcal propertes ca have a well defed measure o a measurable space (R, T. Whe a o empty set M s modelled o such a measure space (R, T, a measure mafold s geerated adoptg above exteded propertes.ths s relevat verfyg the varace of these propertes uder measure varat trasformato o the measure mafold.

11 Iteratoal Joural of dvacemets Research & Techology, Volume 3, Issue 9, September Refereces - [1] Frak Morga (2008 Compactess, Pro.Mathemathca ,ISSN (pp [2] S.C.P. Halakatt ad H. G. Halol(2014 Itroducg the Cocept of Measure Mafold (M 1, T 1, IOSR Joural of Mathematcs(IOSR-JM e-issn ,p- ISSN ,volum-10, Issue 3,ver- II(pp1-11. [3] Todd Esworth ad Peter Nykos(2002 Frst Coutable, Coutably Compact Space ad the Cotum Hypothess J-TMS(Topology ad ppled Mathematcal Scece (pp [4] Yaku Sog ad Hogyg Zhao(2012 O almost Coutably Compact space, Mathemathyka, Bechak 64-2(pp [5] C.Waye Patty (2012 Foudatos of Topology, Secod Edto, Joes ad Bartlett Ida. PVT.LTD New Delh, Caada, Lodo. [6] Ellott H.Leb ad Mchael Loss (2013 alyss, secod Edto,Graduate studes Mathematcs V-14 merca Mathematcal [7] Jho Huter [2011] Measure Theory, Departmet of Mathematcs, Uversty of Calfora at Pars. [8] M.M Rao (2004 Measure Theory ad tegrato, Secod Edto, Revsed ad Expaded. Marcel Dekker.Ic. New York Basel. [9] M.Papadmtraks (2004 Notes o measure theory, Departmet of Mathematcs, Uversty Of Crete utum of [10] Terece Tao (2012 Itroducto to measure theory, merca Mathematcal Socety, Provdece RI.