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Vb’ k Æ9mA d /k %& $1. P f(x) E_uB /k rh#[ Ekr,5E[fr]B \.5E[f=r]B vf(x)h)B ’#℄#[ Ekr,E[fr]B #[ dkα,:{rn}h α T Ek E[fa]=∞Sn=1E[frn], E[frn]B E[fα]hB n f(x)hE_ B /k ℄#[ Ekr,E[f=r]B f(x) hB G\ E=(−∞,∞),zh(−∞,∞) B 5 #[ x∈z,f(x)=√3;x6∈z,f(x)=√2, #[ Ekr,E[f=r]=∅hB %E[f√2]=zh B f hB 2.‘f(x),fn(x)(n=1,2,···)h W;[a,b]_ d/k ku k i ∞\k=1limn→∞E|fn−f|1khE efn(x)jJ f(x) 5 ’#‘AuE fnjJ 5 #[ x∈A,[ k, N,enN |fn(x)−f(x)|1k, x∈limn→∞E|fn−f|1k. k [ | x∈∞\k=1limn→∞E|fn−f|1k.#[ x∈∞Tk=1limn→∞E|fn−f|1k,#[ ǫ0, k0,e1k0ǫ, x∈limn→∞Eh|fn−f|1k0iB N,enN x∈Eh|fn−f|1k0i,7|fn(x)−f(x)|1k0ǫ,n limn→∞fn(x)=f(x),7x∈A.n A=∞\k=1limn→∞E|fn−f|1k.3.‘{fn}uE_B /kK Po jJ 5Æ&^ 5!hB ’# §1E6,limn→∞fn(x)0limn→∞fn(x)!hE_B /k yZE[limn→∞fn=+∞]hfnjJ +∞ n 5 E[limn→∞fn=−∞]hfnjJ −∞ n 5 E[limn→∞fnlimn→∞fn]hfn jJ n 5 fn(x) E_jJ n 5uE−F[limn→∞fn=+∞]−E[limn→∞fn=−∞]−E[limn→∞fnlimfn].1 %hB 5 %&^ 5uE[limn→∞fn=+∞]∪E[limn→∞fn=−∞]∪E[limn→∞fnlimn→∞fn] hB 5 4.‘Eh[0,1] B 5 Mf(x)=(x,x∈E,−x,x∈[0,1]−E.vf(x) [0,1]_h)B |f(x)|h)B !f(x) B ℄0∈E, E[f≥0]=E B ℄06∈E, E[f0]=E B n f(x) B x∈[0,1] |f(x)|=xhI~/k n |f(x)| [0,1]_hB 5.‘fn(x)(n=1,2,···)hE_a.e. z B /kK %|fn|a.e.jJ z/kf. #[ ǫ0 ÆkcÆB 5E0⊂E,m(E\E0)ǫ,e E0_# Tn |fn(x)|≤c. FmE∞.’# p E[|fn|=∞],E[fn6→f]!hL5 n=0,1,2,···.ME1=E[fn6→f]∪(∞Sn=0E[|fn|=∞]), mE1=0. E−E1_fn(x)! z UjJ f(x).ME2=E−E1, [ x∈E2,supn|fn(x)|∞. E2=∞[k=1E2[supn|fn|≤k],E2[supn|fn|≤k]⊂E2[supn|fn|≤k+1].n mE2=limk→∞mE2[supn|fn|≤k]. k0emE2−mE2[supn|fn|≤k0]ǫ.ME0=E2[supn|fn|≤k0],c=k0. E0_ #[ n,|fn(x)|≤c,%m(E−E0)=m(E−E2)+m(E2−E0)ǫ.6.‘f(x)h(−∞,∞)_ I~/k g(x)u[a,b]_ B /k f(g(x))hB /k ’#:E1=(−∞,∞),E2=[a,b]. f(x) E1_I~ -#[ dkc,E1[fc]h {_ 5 ‘E1[fc]=∞Sn=1(αn,βn),S (αn,βn)hS, W;(BQh z+ αnBQu−∞,βnBQu+∞). E2[f(g)c]=∞Sn=1E2[αngβn]=∞Sn=1(E2[gαn]∩E2[gβn]), ug E2_B E2[gαn],E2[gβn]!B -E[f(g)c]B 7.‘/kKfn(x),(n=1,2,···) =5E_”3 _” jJ f(x), P{fn}a.e.jJ f.’# ufn(x) E_”3 _” jJ f(x),n #[ δ0, B 5Eδ⊂E,em(E−Eδ)δ%fn Eδ_ jJ f(x).‘E0hE fn jJ Yq #[ δ,E0⊂E−Eδ( uEδ_fnjJ),n mE0≤m(E−E0)δ,Mδ→0, mE0=0.n fn(x) E_a.e.jJ f(x)( =r).28.i NERE H ’#NERE u f(x)hE_/k #[ δ0, 5Eδ⊂Eef(x) Eδ_hI~/k Um(E−Eδ)δ, f(x)hE_a.e. zB /k #[ 1/n, 5En⊂E,ef(x) En_I~ Um(E−En)1n.ME0=E−∞Sn=1En, #[ n, mE0=m(E−∞Sn=1En)≤m(E−En)1n.Mn→∞, mE0=0.E=(E−E0)∪E0=(∞Sn=1En)∪E0=∞Sn=0En.#[ dka,E[fa]=E0[fa]∪(∞Sn=1En[fa]), f En_I~ B En[fa]B %m∗(E0[fa])≤m∗E0=0,n E0[fa] B %E[fa]hB fhB uf En_ z - ∞Sn=1En_ z n f(x)a.e. z 9.‘/kK{fn} E_ jJ f,Ufn(x)≤g(x)a.e. E,n=1,2,···.i f(x)≤g(x) E_82 H ’# ufn(x)⇒f(x), {fni}⊂{fn},efni(x) E_a.e.jJ f(x).‘E0hfni(x) jJ f(x) 5 En=E[fng]. mE0=0,mEn=0.m(∞Sn=0En)≤∞Pn=0mEn=0. E−∞Sn=0En_fni(x)≤g(x),fni(x)jJ f(x),n f(x)=limfni(x)≤g(x) E−∞Sn=0En_ H 7f(x)≤g(x) E_82 H 10.‘ E_fn(x)⇒f(x),Ufn(x)≤fn+1(x)82 H n=1,2,···, 82 fn(x)jJ f(x).’# ufn(x)⇒f(x), {fni}⊂{fn},efni(x) E_a.e.jJ f(x).‘E0hfni(x) jJ f(x) 5 En=E[fnfn+1], mE0=0,mEn=0. m(∞[n=0En)≤∞Xn=0mEn=0 E−∞Sn=0En_ fni(x)jJ f(x),Ufn(x)h fn(x)jJ f(x).( }K KjJ }K ajJ s 4z).7 X +L5∞Sn=0Ent fn(x)jJ f(x),?hfn(x)a.e.jJ f(x).11.‘ E_fn(x)⇒f(x),%fn(x)=gn(x)a.e. H n=1,2,···, gn(x)⇒f(x).’#‘En=E[fn6=gn] m(∞Sn=1En)≤∞Pn=1mEn=0.#[ σ0,E[|f−gn|≥σ]⊂(∞Sn=1En)∪E[|f−fn|≥σ].n mE[|f−gn|≥σ]≤m(∞[n=1En)+mE[|f−fn|≥σ]=mE[|f−fn|≥σ].3 ufn(x)⇒f(x),n 0≤limmE[|f−gn|≥σ]≤limmE[|f−fn|]≥σ=07gn(x)⇒f(x).12.‘mE+∞, P E_fn(x)⇒f(x) rh # {fn} [1 /kK{fnk}, {fnk} /kK{fnkj},e limj→∞fnkj(x)=f(x),a.e. E.’# | DlEH7B w (| ℄{fn(x)} E_ jJ f(x). η00,e kK{mE[|fn−f|≥η0]} jJ L - kǫ00, 6 /kK{fnk},e mE[|fnk−f|≥η0]ǫ00.(1) /kK{fnk} 82 jJ f(x) /kK gd_ ℄ /kK{fnkj} E_a.e.jJ f, mE+∞, C *E E_fnkj⇒f(x), Æ(1)fO$ 13.‘mE∞,82 z B /kKfn(x)0gn(x),n=1,2,···,( jJ f(x)0g(x), P (1)fn(x)gn(x)⇒f(x)g(x);(2)fn(x)+gn(x)⇒f(x)+g(x);(3)min{fn(x),gn(x)}⇒min{f(x),g(x)};max{fn(x),gn(x)}⇒max{f(x),g(x)}.’#(1) f(x)a.e. z n mE[|f|=∞]=0. ∞Tn=0E[|f|≥n]=E[|f|=∞],UE[|f|≥n]⊃E[|f|≥n+1]0E[|f|≥1]⊂E,mE[|f|≥1]≤mE∞,n mE[|f|=∞]=limn→∞mE[|f|≥n]=0.sElimn→∞mE[|g|≥n]=0.#[ ǫ0,σ0, k,mE[|f|≥k]ǫ50mE[|g|≥k]ǫ5s H Mσ0=minσ2(k+1),1, fn⇒f,gn⇒g, N,enN mE[|gn−g|≥σ0]ǫ5,mE[|fn−f|≥σ0]ǫ5s H E[|gn|≥k+1]⊂E[|g|≥k]∪E[|gn−g|≥1]⊂E[|g|≥k]∪E[|gn−g|≥σ0].mE[|gn|≥k+1]≤mE[|g|≥k]+mE[|gn−g|≥σ0]ǫ5+ǫ5=2ǫ5. Eh|gnfn−gnf|≥σ2i⊂E[|gn|≥k+1]∪E|fn−f|≥σ2(k+1)⊂E[|gn|≥k+1]∪E[|fn−f|≥σ0].n mEh|gnfn−gnf|≥σ2i≤mE[|gn|≥k+1]+mE[|fn−f|≥σ0]2ǫ5+ǫ5=3ǫ5.%Eh|fgn−fg|≥σ2i⊂E[|f|≥k+1]∪E|gn−g|≥σ2(k+1)⊂E[|f|≥k]∪E[|gn−g|≥σ0].4n mEh|fgn−fg|≥σ2i≤mE[|f|≥k]+mE[|gn−g|≥σ0]ǫ5+ǫ5=2ǫ5. E[|gnfn−gf|≥σ]⊂Eh|gnfn−gnf|≥σ2i∪Eh|fgn−fg|≥σ2i,n mE[|gnfn−gf|≥σ]≤mEh|gnfn−gnf|≥σ2i+mEh|fgn−fg|≥σ2i3ǫ5+2ǫ5=ǫ.7#[ ǫ0,σ0, N,enN mE[|gnfn−gf|≥σ]ǫ,n gnfn⇒gf.(2)E[|(fn+gn)−(f+g)|≥σ]⊂Eh|fn−f|≥σ2i∪Eh|gn−g|≥σ2in mE[|(fn+gn)−(f+g)|≥σ]≤mEh|fn−f|≥σ2i+mEh|gn−g|≥σ2i,limn→∞mE[|(fn+gn)−(f+g)|≥σ]≤limn→∞mEh|fn−f|≥σ2i+limn→∞mEh|gn−g|≥σ2i.7fn+gn⇒f+g.(3)x ℄fn⇒f, |fn|⇒|f|.gd_ E[|fn−f|≥σ]⊃E[||fn|−|f||≥σ].n limn→∞mE[||fn|−|f||≥σ]≤limn→∞mE[|fn−f|≥σ]=0,7|fn|⇒|f|. ℄fn⇒f,#[ a6=0,afn⇒af.gd_ E[|afn−af|≥σ]=E|fn−f|≥σ|a|,n limn→∞mE[|afn−af|≥σ]=limn→∞mE|f