EXTENSION OF AN ADDITIVE FUNCTIONS NUMARABILE

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1 EXTENSION OF AN ADDITIVE FUNCTIONS NUMARABILE ProfDrd Bogda P Costati Uiversitatea di Craiova AsuivPecigia Olimpia Uiversitatea Costati Bracusi,Tg-jiu ABSTRACT: All measures to start the costructio of theories from user a fudametal theorem of a Lebesgue whose cotemporary formulatios is due to c Caratheodory NORMED ALGEBRAS Lebesgue Theorem Caratheodory-Demostratio Either o AB completewe cosider subalgebra with a quasimasura We will ow udertae the followig tass: costructio of a subalgebras regular that icludes ad a quasimasura additive umarabila o so, where it is possible, for ay I order to accomplish the tas, we will defie a outer quasimasura by associatig with each the crowd S that is all most families umarabile satisfyig the coditio sup, ad by: if S The Demostratio Either a AB completewe cosider subalgebra with a quasimasura We will ow udertae the followig tass: costructio of a subalgebre regular that icludes ad a quasimasura additive umarabila o so, where possible, for ay I order to accomplish the tas, we will defie a outer quasimasura by associatig with each the crowd S that is all most families umarabile satisfyig the coditio sup, ad by: if S We ote the mai characteristics of this eteral quasimasuri 1 If the This property is obviousimplies that is fiitethe followig property is also evidece 2 Outside measuremet este mootoa: ivolve 3 If the 247 Fiabilitate si Durabilitate - Fiability & Durability No 1/ 214 Editura Academica Brâcuşi, Târgu Jiu, ISSN X

2 Cosiderig that the measure is totally additive, 248 Ideed, ivolve establishig a chace, associated with each family S so that the iequality remaisclearly, geder equality 2 belogs to S ; ad, that's why Because is radom, we obtai the iequality Now we arrage the so equality ( u ) ( u C ) ( u) (4) with all the elemets eep for ay u It is clear that always ( u ) ( u C) ( u), so we o loger stays tha to validate the seemig iequality bacside Z 1,2 si be the sequece Z What teds to repeat z The Lemma 2Either ( u z) lim ( u z ) for ay u Demostratio Everythig is obvious z decreasessuppose that z complete all the steps, admits metric spaces, are homeomorfe to each other ( u z) u z1 z u z 1Cz u z u z 1 lim uz Lemma is proved Lemma 3 The Crowd Demostratio I coditio of the mai (4) items si is a subalgebra o regular basis C have equal status, so z ad z1 The cotais each elemet together with the complemetarul or Is Yes y, ad we have z y cu u We will have ( u) ( u ) ( u C ) ( u y) ( u C ) ( u C y) ( u C C ) ( u y) y ( u Cy ) ( u C y) ( u C Cy) y ( u z) ( u C z ), Fiabilitate si Durabilitate - Fiability & Durability No 1/ 214 Editura Academica Brâcuşi, Târgu Jiu, ISSN X

3 Sice ( u C ) ( u C ) ( u C C ) u C ( C ) ( C C) u C u C Thus z Note that is the subalgebra We remai to show that is o a regular basiseither,,,, a 1 2 m umarabila ad submultime yi im yi m im m yi i im im y Cosiderig that so far as is ( o ) - cotiue: I the previous Lemma, ( u ) lim ( u y ), ( u C )lim ( u C ) y Hece, ( u ) ( u C ) ( u) Thus, It has bee demostrated that is a subalgebra o regular basis Lemma 4 Demostratio Either Choose a item u ad a correspodet by chace Su Families of elemets of the form z z ad C z the correspodig elemets are u ad that C u So z z z C z z () z z z C z ( u) ( C u) Taig ito accout that by chace, we coclude that ( u) ( u) ( C u) Thus, Aioa is demostrated Lemma 5 If ( ) the DemostratioSe Yes u, as a result ( u) ( u) ( C u) ( o) i z 249 Fiabilitate si Durabilitate - Fiability & Durability No 1/ 214 Editura Academica Brâcuşi, Târgu Jiu, ISSN X

4 I fact, the coicidece of topology ( o ) ad ( os ) It is sufficiet to refer la Theorem 4 Suppose that y, ad d y Atuci y y y y C ( ) ( y) y So, see for eample aditivitatiiso, for ay sequece disjucta i, 1 We have m m 1 ad thus: 1 1 m 1 Cosiderig the iequality: 1 m m 1 1 1, which is maitaied, we get to equality: lim i y if i yi yi mi im m, (4) im The right side of this iequalities disappears: They are studyig the structure of subalgebrei that is defied quasimasura Se da u Icidetally, we choose the correspodig S 1,2, so 25 Fiabilitate si Durabilitate - Fiability & Durability No 1/ 214 Editura Academica Brâcuşi, Târgu Jiu, ISSN X

5 1 ( y) y 1,2, to be valid Now we will put sup It is clear that ad belogs to subalgebrei ad also sup 1 ( y), So, Thus, each elemet is represeted as u, where, u, si Lemma 1, 1 ( o) So, we have Eve more so, sice the is there a submultime umarabil umarabila such 2 sup, as: where: sup sup 2 Now we ca cosider sup a substitute for Therefore, 2 2 Lemma is proved Defiitio: a quasimasura v a subalgebra It is called complete as log as the coditios,, ad v ivolve (ad obviously v ) As well as i Lemma 6, quasimasura I built it is complete Lemma 9 Either a additive umarabila full qasimasura to some subalgebra - that icludes regular for ay The: 1) cosidered as a metric space What else ca we say about the properties i this space? For eample, will be coected? The aswer is egative, i geeral, sice some lie algebras are ormed listed all fiite AB, which are clearly discoected It is assumed that 251 Fiabilitate si Durabilitate - Fiability & Durability No 1/ 214 Editura Academica Brâcuşi, Târgu Jiu, ISSN X

6 Teorema 4 A ABN is a metric space cotiue Astfel, result, Applyig this to a elemet It remais to prove the Lemma eut first First we will show that if v the v ad v satisfyig v v elemet, v v his So v ad v v Ideed, as i Lemma 8, There is a All items belog to him ad thus Quasimasura It is thus complete; v ad Now it is clear that each belog to him, taig ito accout that ca be writte as u, where It is a elemet ad u So iclusiveess ad Juliet are demostrated We eed complete the demostratio oly poit 1) Lemma 1 If sup ( y) y I geeral: the regularity of the algebra Thus, each ca be cosidered a ABN space metric What else ca we say about the properties i this space? For eample, will be coected? The aswer is egative, i geeral, sice some lie algebras are ormed listed all fiite AB, which are clearly discoected Ideed, i this case for ay, so the 1 of a eteral steps leads to the desired equality Teorema 6 Either a AB complete, ad a subalgebra of, ad a quasimasura i The there is a subalgebra) regular icludig ad (b)) a quasimasura additive umarabila i with the followig properties: 1) for ay ; 2) for each additive umarabila quasimasura i ad for ay Image chec It is true for ay, the iequality How is umarabila the the additive equality It is true for ay i other words It is a etesio of Call quasimasura It is built i the proof theorem Fiabilitate si Durabilitate - Fiability & Durability No 1/ 214 Editura Academica Brâcuşi, Târgu Jiu, ISSN X

7 CONCLUSIONS The mai issue is topical because it approaches the fuzzy systems uderlyig the artificial itelligece that is implemeted i the ecoomic ad idustrial machies REFERENCES [1] Balbes si Dwiger,The curatours of the Uiversity of Missouri,1974 [2] Dumitru Buşeag,Categories of Algebric Logic, Editura Academiei 253 Fiabilitate si Durabilitate - Fiability & Durability No 1/ 214 Editura Academica Brâcuşi, Târgu Jiu, ISSN X