σ-anant mojmap

ganitat, jarr "x" sanchachya upasanchanchya "σ"-bijganit σ varr paribhashit kelele sakaratmak (kinwa swakshari kelele) mojmap "μ" heey maryadit mojmap mhatale jaate, jarr "μ" ("x") hii eka maryadit vastavik sankhya asel (φ aivaji) aanee σ madheel sanch "a" haah maryadit mojamapacha asel, jarr "" μ "(" a ") <φ"."jarr" "x" "heey mojata yenyajogya sanchanche, pratyek sanch maryadit mojmapasah, mojata yenyajogya sanyojanache asel, tarr" "μ" "mojamashala σ-anant mhanatat". mojmap jaaget asalelya sanchala 'σ'-anant mojmap asey mhatale jaate, jarr too mojata yenyajogya sanchancha maryadit mojmapasah ganata yenyajogya sanyog asel. eka mojmap σ-anant asane hii maryadit asanyapeksha kamkuvat aut aahe, i.eae. serve maryadit mojmap σ-anant aahet parantu (anek) σ-anant mojmap aahet jeey maryadit naheet.
eka vegali parantu sambandhit kalpana jee σ-parichitateshi gondhaloon jaau naye tee mhanje s-parichitata.
vyakhya.
sootra _ 1 heey mojata yenyajoge sthan aanee sootra _ 2 heey tyavar mojmap asoo dya.
jarr teey khaleel chaar samatulya nikashanpaiki eka purn karat asel tarr mojamapachya sutrala σ-anant mojmap mhanatatah
jarr sootra _ 2 heey sootra _ 27-anant mojmap asel, tarr mojmap antaral sootra _ 28 laa sootra _ 29-anant mojmap jaaga mhanatat.
udaharane.
lebsgue mojmap.
udaharanarth, vastavik sankhyanvaril lebsgue mojmap maryadit nahee, parantu teey σ-anant aahe. kharokharach, serve purnankansathi antarancha vichar qaraa; asey anek antar aahet, pratyekache mojmap 1 aahe aanee tyanche millen hii sampoorn vastavik resha aahe.
mojmap.
vaikalpikaritya, mojanichya mojmapasah vastavik sankhyancha vichar qaraa; konatyahi maryadit sanchache mojmap heey sanchatil ghatakanchi sankhya asate aanee konatyahi anant sanchache mojmap anant asate. heey mojmap "σ"-anant nahee, kaaran maryadit mojmap asalelya pratyek sanchamadhye keval maryaditpane anek bindu asatat aanee sampoorn vastavik resha jhakanyasathi asha anek sanchanna aganit vel lagel. parantu, mojanichya mojmapasah naisargik sankhyanchya sootra _ 30 chaa sanch "σ"-anant aahe.
sthanik patalivar sankshipt gut.
sthanik patalivar sankshipt gut jeey σ-sankshipt aahet teey haar mojmapasah σ-anant aahet. udaharanarth, serve jodalele, sthanik patalivar sankshipt gut "jee" heey σ-sankshipt aahet. heey pahanyasathi, 'v' haah tulanene sankshipt, samamiteey (mhanje 'v' = 'v'-1) olakh ughadanara parisar asoo dya. mug
haah 'jee' chaa khula upasamuh aahe. mhanoon 'etch' dekhil bund aahe kaaran tyache purak heey khulya sanchanche eka sanyojan aahe aanee 'jee' chya jodanidware, 'jee' swatach asane aavashyak aahe. asha prakare serve jodalele khote gut haar mojmapasah σ-anant asatat.
ekahi namuna nahee.
keval 0 aanee sootra _ 32 hii done moolye ghetalele konatehi kshullak nasalele mojmap spashtapane σ-anant nahee. sootra _ 33 madheel eka udaharan aaheh serve sutransathi _ 34, sootra _ 35 jarr aanee fakt jarr a rikame nasel tarr; dusare mhanjeh serve sutransathi _ 34, sootra _ 35 jarr aanee fakt jarr a aganit asel tarr, anyatha 0. yogayogane, donhi anuvad-bhinn aahet.
malmatta.
σ-anant mojmapanmadhil vargamadhye kahee atishay soyiskar gunadharm aahet; yaa sandarbhat σ-parimanachi tulana topological spaceschya pruthakkarankshamateshi kelee jaau shakate. vishleshanatil kahee prameyanna gruhitak mhanoon σ-parimap aavashyak asate. samanyatah, redon-nicodime pramey aanee fubiniche pramey heey donhi samavisht asalelya upayanvar σ-parimanachya gruhitkamadhye namood kele jaatat. tathapi, segalchya peparmadhye darshavilyapramane "mojmap jaganchi samatulyata" ("aii. emm.". jeey. ganeet."73,275 (1953)) tyanna keval kamkuvat sthitichi aavashyakta asate, mhanje" "sthanikikaranakshamata". "
jaree 'σ'-anant nasalele mojmap kadhikadhi pathological manale jaatat, parantu teey pratyakshat agadi naisargikaritya ghadatat. udaharanarth, jarr 'x' heey 'r' yaa aakarmanache 'hasdorf' chey metrik space asel, tarr 'x' vareel mojmap manale tarr serve qamee-aayami hasdorf mojmap heey 'σ'-anant nasatat.
sambhavyata mojamapachi samanata.
x" yaa jagevaril konatehi σ-anant mojmap "" m "heey" x "vareel sambhavyata mojamapachya samtulyah" v "" n "," n "ωn, heey" x "chey aavaran asoo dya, joou jodine visangat karoon mojata yenyajoge maryadit" "m"-mojmap sanch asoo dya aanee "" w "" n "," n "ωn, haah sakaratmak sankhyancha (vajan) kram asoo dya, mhanje qee
ν" dware paribhashit kelele mojmap
tyanantar 'x' varr 'μ' sarkhyach achookpane null sanchansah sambhavyateche mojmap aahe.
sambandhit sankalpana.
madhyam upay.
jarr serve sutransathi _ 44 aanee sutrasathi _ 45 sutrasah, jastit jaast mojmap karnyasarkhe anek open seet sootra _ 42 asateel tarr borel mojmap (borel sootra _ 27-bijganitavaril sthanik maryadit mojamapachya arthane) sootra _ 2 laa madhyam mojmap mhanatat.
pratyek madhyam mojmap heey eka sootra aahe-29-anant mojmap, ulat kharey nahee.
vighatansheel upay.
eka mojamapala vighatansheel mojmap mhanatat, serve sutransathi _ 43 sutrasah vighatit mojata yenyajoge sanch sootra _ 47 aanee sootra _ 50 aahet. vighatanaksham mojmapansathi, maryadit mojmapasah mojata yenyajoge sanch sankhyevar konatehi bandhan nahee.
pratyek sootra _ 29-anant mojmap heey vighatit karnyayogya mojmap aahe, ulat kharey nahee.
ess-anant upay.
jarr mojmap sootra _ 2 heey jastit jaast mojmap karnyayogya anek maryadit mojamapanchi berries asel tarr tyala ess-finite mojmap mhanatat.
pratyek σ-anant mojmap s-anant aahe, ulat kharey nahee. purava aanee prati-udaharanasathi s-infinite measure#relation teey σ-infinite mojmap pahaa.

