redon-nicodime pramey

ganitamadhye, redon-nicodime pramey haah mojmap siddhantacha parinam aahe joou samaan mojata yenyajogya jaaget paribhashit kelelya done mojmapanmadhil sambandh vyakt karato. mapan" heey eka sanch karya aahe jeey mojata yenyajogya jagechya mojata yenyajogya upasanchanna susangat pariman dete. mojamapachya udaharananmadhye kshetra aanee aakarman yancha samavesh hoto, jethe upasanch heey bindunche sanch asatat; kinwa ekhadya ghatanechi sambhavyata, jee vyapak sambhavyata jaaget sambhavya parinamancha upasanch asate.
aadhich dilelya eka mojamapatoon naveen mojmap midvanyacha eka marg mhanje jagechya pratyek bindula ghanata niyukt karane, nantar vyaj mojata yenyajogya upasanchavar samakalit karane. heey asey vyakt kele jaau shakate
konatyahi mojata yenyajogya upasanchasathi naveen mojmap kuthe paribhashit kele jaat aahe aanee karya heey dilelya binduvaril ghanata aahe. avibhajya heey vidyaman mojamapachya sandarbhat aahe, jeey bahutekda vastavik reshevaril pramanit lebsg mojmap kinwa "aine"-aayami euclidean space (lambi, kshetrafal aanee aakarmanachya aaplya manak kalpananshi sambandhit) asoo shakate. udaharanarth, jarr vastuman ghanateche pratinidhitva kele aanee tri-aayami jaaget teey lebsgue mojmap asel, tarr sthanik pradeshatil ekoon vastuman samaan asel.
redon-nicodime pramey moolatah asey sangato qee, kahee vishisht paristhitinmadhye, konatehi mojmap tyach jaaget mojamapatoon mojamapachya sandarbhat asha prakare vyakt kele jaau shakate. tyanantar yaa karyala redon-nicodime vyutpann mhanatat aanee sootra _ 2 dware darshavile jaate. sambhavyata siddhantat eka mahattvacha anuprayog aahe, jyamule yadruchchik chull sambhavyata ghanata karya hote.
yaa prameyala johann redonche naav denyat aale aahe, jyane 1913 madhye antarnihit jaaga asalelya vishesh prakaranasathi pramey siddha kele aanee 1930 madhye samanya prakaran siddha siddha oto nikodiyamsathi. 1936 madhye hans frydenthalne frydenthal spectral pramey siddha karoon redon-nicodium pramey samanyikrut kele, joou eriz space siddhantatil parinam aahe; yaat eka vishesh prakaran mhanoon redon-nicodium pramey aahe.
jarr redon-nicodime pramey samanyikaranat dekhil maan asalelya karyansathi 'mutatis mutandis' asey mhatale gele tarr banach spacemadhye redon-nicodime gunadharm asalyache mhatale jaate. serve hilbert jaganmadhye redon-nicodime gunadharm asato.
aupcharik varnan.
redon-nicodime pramey.
redon-nicodime prameyamadhye mojata yenyajoge avkash sootra _ 3 samavisht aahe jyavar done σ-anant mojmap paribhashit kele aahet, sootra _ 4 aanee sootra _ 5 
tyat asey mhatale aahe qee, jarr sootra _ 6 (mhanje, sootra _ 7 heey sootra _ 4 chya sandarbhat purnapane akhand asel), tarr eka sootra _ 9-mojata yenyajoge karya sootra _ 10 astitvat aahe, jenekarun konatyahi mojata yenyajoge sootra _ 11 sathee
sootra _ 12
redon-nicodime vyutpann.
vareel samanateche samadhan karnare karya sootra _ 13 mhanje, jarr sootra _ 14 heey dusare karya asel jeey samaan gunadharm purn karate, tarr sootra _ 15. karya sootra _ 13 heey samanyatah sootra _ 17 asey lihile jaate aanee tyala sootra mhanatat. sanketachi nivad aanee karyache naav heey heey vastav pratibimbit karate qee heey karya gananat vyutpannasarkhe aahe, yaa arthane qee teey eka mojamapachya mojamapatoon mojamapachya sandarbhat ghanatechya badalachya darache varnan karate (jyaa prakare jacobian nirdharak bahuvidh ektrikaranat vaparla jaato).
swakshari kelelya kinwa guntaguntichya upayayojanancha vistar.
swakshari kelelya aanee guntaguntichya upayansathi samaan pramey siddha kele jaau shakateh mhanje, jarr sootra 4 heey eka nakaratmak nasalele σ-anant mojmap asel aanee sootra 7 heey eka maryadit-moolya asalele swakshari kelele kinwa jatil mojmap asel, jase qee sootra 20 mhanje sootra 7 heey sootra 22 chya sandarbhat purnapane akhand asel tarr sootra 25 varr eka sootra _ 4-samakalit vastavik-kinwa jatil-moolya asalele karya sootra _ 14 aahe, jenekarun pratyek mojata yenyajogya sanch sutrasathi _ 26
sootra _ 27
udaharane.
khaleel udaharananmadhye, sanch haah vastavik madhyantar [0,1] aahe aanee sootra _ 9 heey borel sigma-bijganit aahe.
anuprayog.
sambhavyata siddhant.
sambhavyata siddhantachya kalpana sambhavyata vastuman aanee vastavik sankhyanvar paribhashit sambhavyata ghanatepasun aniyantrit sanchanvar paribhashit sambhavyata upayanparyant vistaranyasathi pramey khoop mahatvache aahe. eka sambhavyatechya mojamapatoon mojamapatoon sambhavyatechya mojamapat badal karane shakya aahe qaa aanee kase heey teey sangate. visheshtah, yadruchchik chalche sambhavyata ghanata karya heey kahee moolbhut mojamapachya (samanyatah satat yadruchchik chalansathi lebsgue mojmap) sandarbhat prerit mojamapache redon-nicodime vyutpann aahe.
udaharanarth, sambhavyata upayansathi sashart apekshanche astitva siddha karnyasathi yacha vapar keela jaau shakato. nantarchi sankalpana hii sambhavyata siddhantatil eka pramukh sankalpana aahe, kaaran sashart sambhavyata hii fakt eka vishesh baab aahe.
aarthik ganeet.
itar kshetranmadhye, aarthik ganeet heey, visheshtah girsanov prameyadvare, yaa prameyala mothya pramanat vaparte. sambhavyatechya mojamapache asey badal heey vyutpannanchya tarksangat kimmatincha paya aahet aanee vastavik sambhavyateche jokhim tatasth sambhavyatemadhye rupantar karnyasathi vaparale jaatat.
mahititil farak.
jarr "maay" aanee "maay" mojmap sampale asateel, aanee "maay" maay "ν"
σ-parimanipanachi dharana.
vareel redon-nicodime pramey asey gruheet dharato qee jyachya sandarbhat "ν" chya badalachya darachi ganana kelee jaate teey mojmap "μ" heey σ-anant aahe.
nakaratmak udaharan.
yethe eka udaharan aahe jevha "maay" heey σ-anant nasate aanee redon-nicodime pramey tikavanyat ayashasvi hote.
vastavik reshevaril borel σ-bijganitacha vichar qaraa. borel sanchatil mojamapache mojmap, jarr maryadit asel tarr ghatakanchi sankhya mhanoon paribhashit kele jaau dya, aanee anyatha. heey kharokharach eka upay aahe heey konihi tapasu shakate. pratyek borel sanch haah jastit jaast maryadit sanchancha gananiy sangh nasato, tyamule too-anant nasato. yaa borel bijganitevar nehamiche lebsgue mojmap asoo dya. mug, yaa sandarbhat purnapane satatyapurn aahe, kaaran eka sanchasathi fakt rikama sanch asel tarach asato aanee nantar too dekhil shoonya asato.
asey gruheet dhara qee redon-nicodime pramey dharan karato, mhanje kahee mojata yenyajogya karyasathi
serve boral sanchansathi. eka singleton sanch mhanoon ghetalyas, aanee vareel samanatecha vapar karoon, ekhadyala aadhalate
serve vastavik sankhyansathi. heey soochit karate qee karya, aanee mhanoon lebsgue mojmap, shoonya aahe, joou eka virodhabhas aahe.
sakaratmak parinam.
sootra _ 58 gruheet dharalyas, jarr sootra _ 4 sthanikaritya asel aanee sootra _ 7 heey sootra _ 4, i chya sandarbhat "sulabh" asel tarr redon-nicodime pramey dekhil dharan karato.eae., serve sutransathi sootra _ 62 _ 63
purava.
haah vibhag yaa siddhantacha eka mojmap-saiddhantik purava deto. hilbert space paddhati vaparoon eka karyatmak-vishleshanatmak purava dekhil aahe, joou pratham wonn newmanne deela hota.
maryadit upayansathi aanee, karyancha vichar karane hii kalpana aahe. asha serve karyanche sarvoch, monotone abhisaran prameyasah, nantar redon-nicodime vyutpann pradan karate. maryadit upayambaddalachya tantrik tathyachya anusaranachya sandarbhat urvarit bhag ekvachan aahe hii vastusthiti. ekada parinam maryadit upayansathi sthapit jhaala qee,-anant, swakshari kelele aanee jatil upay naisargikaritya kele jaau shakatat. tapasheel khalee deela aahe.
maryadit upayansathi.
aadhi vistarit-moolya asalele umedvar tayar karane, samaja aanee teey donhi maryadit-moolya asalele nakaratmak nasalele upay aahet. tyaa vistarit-moolya mojata yenyajogya karyancha sanch asoo dya, jase keeh
, kaaran tyat kiman shoonya karya asate. aata samaja, aanee samaja haah eka aniyantrit mojata yenyajoga sanch aahe aanee tyachi vyakhya karah
mug eka
aanee mhanoon.
aata, asha prakare karyancha kram asoo dya qee
pahilya functionschya jastit jaast sankhyes badaloon, kram vadhat aahe asey gruheet dharale jaau shakate. asey paribhashit kelele vistarit-moolya asalele karya asoo dya
lebsgechya monotone abhisaran prameyanusar, eka aahe
pratyekasathi, aanee mhanoon. tasech, bandhkamadware,
aata samanata siddha karane, kaaran,
eka nakaratmak nasalela upay paribhashit karato. samanata siddha karnyasathi, aamhi teey dakhavato.
samaja; tarr, teey maryadit asalyane, asey eka aahe. yavarun virodhabhas midvanyasathi, aapan swakshari kelelya mojamapasathi sakaratmak sanch shodhato (i.eae. eka mojata yenyajoga sanch, jyachya mojata yenyajogi serve upasanchanmadhye nakaratmak nasaleli mojmap asate), jethe sakaratmak-mojmap dekhil asate. sankalpanatmakdrushtya, aamhi eka sanch shodhat aahot, jithe pratyek bhagat. swakshari kelelya mojamapasathi hahn vighatan vaparane haah eka soyiskar marg aahe.
mug lakshat ghya qee pratyekasathi aahe, aanee mhanoon,
yache soochak karya kuthe aahe? tasech, lakshat ghya qee ichchit asalyas; jarr, tarr (kaaran yaa sandarbhat purnapane satatyapurn aahe), tarr aanee
tyaa vastusthitila virodh karane.
tevhapasunahi
 aanee samadhan milate
heey ashakya aahe kaaran teey sarvochatechya vyakhyeche ullanghan karate; mhanoon, prarambhik dharana jee khotee asali pahije. mhanoon, ichchit mhanoon.
aata maryadit mulyanparyant maryadit, kaaran-ekatmik asalyane, sanch-shoonya aahe. mhanoon, jarr aaa ashee paribhashit kelee gelee aahe
tyanantar ichchit gunadharm aahet.
advitiyatebaddal, mojata yenyajogi karye samadhankarak asoo dya
pratyek mojata yenyajogya sanchasathi. mug, ekatmik aahe, aanee
visheshtah, kinwa. tyanantar asey
aanee mhanoon, teey-jawaljawal sarvatra; tech kharey aahe, aanee asha prakare-jawaljawal sarvatra, ichchit mhanoon.
- anant sakaratmak upayansathi.
jarr aanee teey-anant asateel, tarr teey vichchhedit sanchanchya kramache sanyojan mhanoon lihile jaau shakate, jyapaiki pratyekas aanee donhi antargat maryadit mojmap aahe. pratyekasathi, maryadit prakarnanusar, eka-mojata yenyajoge karya asey aahe qee
pratyek mojata yenyajoga upasanch. tyaa functionsche berries sootra _ 81 heey nantar aavashyak karya aahe jase qee sootra _ 82. 
advitiyatebaddal, pratyek jawaljawal sarvatra adviteey asalyane, tasech aahe.
swakshari kelelya aanee guntaguntichya upayayojanansathi.
jarr-maryadit swakshari kelele mojmap asel, tarr teey jahon-jorden mhanoon vighatit kele jaau shakate, jethe eka mojmap maryadit aahe. tyaa done upayanna mageel parinam lagoo karoon, ekala done karye midatat, jyamule redon-nicodimeche pramey anukrame aanee tyache samadhan hote, jyapaiki kiman eka-samakalit (i.eae., yachya sandarbhat tyache avibhajya maryadit aahe). tevha heey spasht hote qee, advitiyatesaha aavashyak gunadharmanche samadhan hote, kaaran donhi adviteey aahet aanee jawaljawal sarvatra samaan aahet.
jarr eka jatil upay asel tarr too asha prakare vighatit keela jaau shakato, jethe donhi aanee maryadit-moolya asalelya swakshari kelelya upayayojana aahet. vareel yuktivad lagoo kelyane, ekhadyala done karye midatat, jyamule anukrame aanee sathee aavashyak gunadharm purn hotat. spashtapane, heey aavashyak karya aahe.
lebsgu apghatan pramey.
lebsgeche vighatan pramey darshavite qee redon-nicodime prameyatil gruhitak adhik samanya samanya paristhititahi aadhalu shakatat. konatehi paripoorn satatya gruheet naan dharata, avkash sootra 84 mojamapavar σ-anant sakaratmak mojamapache sootra 4 aanee sootra 9 varr σ-anant swakshari kelele mojamapache sootra 7 vicharat ghya. tyanantar sootra 9 varr adviteey swakshari kelele mojmap sootra 87 aanee sootra 88 astitvat aahet, jase qee sootra 90, sootra 91 aanee sootra 92. tyanantar jodee sootra 93 varr redon-nicodime pramey lagoo kele jaau shakate.

