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REAL ANALYSIS LECTURE NOTES - Atlanta, GA

MeasureandpointwiseconvergenceAlthoughco nvergencein measuredoes notimplypointwiseconvergence,we dohave thefollowingweaker (butstillveryuseful) !f, thenthereexistsa subsequenceffnkgk2 Nsuch thatfnk! !f, we can ndn1< n2< such that8n nk; njf fnj>1ko 12k:De neEk=njf fnkj>1koandHm=1Sk=mEk:Thenwe have (Ek)<12kand (Hm) 1Xk=m12k=12m 1:SetZ=1Tm=1Hm:Then (Z) (Hm) 1=2m 1foreverym, so we have (Z) = =2Z, thenx =2 Hmforsomem. Hencex =2 Ekforallk m, which impliesjf(x) fnk(x)j 1k;allk m:Thusfnk(x)!f(x) forallx =2Z. SinceZhasmeasurezero,we thereforehave pointwiseconvergenceoffnktofalmosteverywhere. Asanimportant specialcasewe have !finL1(X), thenthereexistsa subsequenceffnkgk2 Nsuch thatfnk! andexpandonthetext\RealAnalysis:ModernTechniquesandtheirApplications,"2nded.,by MODESOF Cauchycriterionfor convergencein measureAlthoughconvergencein measureis notassociatedwitha particularnorm,thereis stillausefulCauchy criterionforconvergencein , we say thatffngn2 ZisCauchyin measureif8" >0; fjfm fnj "g!

REAL ANALYSIS LECTURE NOTES: 2.4 MODES OF CONVERGENCE CHRISTOPHER HEIL 2.4.1 The relation between convergence in measure and pointwise convergence

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