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rih 2(H G _T HeFuliSchoolofMathematicsScienceandComputingTechnologyCentralSouthUniversityC 0( Æ $ &June,2010};iA_ }5‘Xb \ 1§1.1℄K^PwÆ..................................1§1.2℄K Y ...................................4§1.3 ℄K.....................................5§1.4\ /Y℄..................................7(3 .........................................8Fn%tgE‘12§2.1 I U{ Bolzano−Weierstrass ...........12§2.2 ℄ R℄ %M℄..............................13§2.3V) ℄ R℄ %M℄ 7{.......................15§2.4 ℄i ..................................16(3 .........................................18 6Cx~19§3.1$b ......................................19§3.2 b℄K.....................................20§3.3 ℄ bJ..................................22 r6T 24§4.1 bF ℄^Pj J ........................242E 13§4.2Egoroff ..................................29§4.3 bF y7Lusin .........................29§4.4Wb ~’...................................30(3 .........................................31*^Jx~32§5.1#)F [$................................33§5.2 [F .....................................37§5.3Fubini ...................................42§5.4&$l\ [$................................44(3 .........................................45*1...........................................474p’050?6AaY ℄ zvWF 4 fLT C ℄$1 R F J %( M O~}l*℄K4 Ys w O~ T}l*℄K ℄K wÆ ℄K Y ℄ Ie ℄K $zlK[| ℄K i4Y^7{E℄K T $zIK[ Y C℄K wÆ v9 fjzv$1 fHV+F \:4 T(V+℄K ℄K4 YO~}l*℄+ j\:I6\: ,L Pob 2 9 ℄^PYz T (/℄KY X|2 ℄n p#(℄K i m X|!dBernstein n p#(/ m U Y!dCC J (f / ℄ [℄da| ℄ ) O~ |) I3 vr* C! #N Pob+ 2 ℄ J [J\$ ℄I\ ℄ §1.1‘Xb = 1-1-1 (B−A)∪A=B mT o|A⊂B. STJ8a {vjA(B ℄ _ A⊂B.m$J fA⊂B^B−A⊂B Y(B−A)∪A⊂B.if(B−A)∪A⊃BMj# j| Æ1-1-2 A−B=A∩BC. ∀x∈A−B _x∈AWx∈B [_x∈AWx∈BC j|x∈A∩BC.yx∈A∩BC _x∈AWx∈BC.[_x∈AWx∈B V |x∈A−B. j 1 V~℄K^PY 2 Æ1-1-3 4 (3) (4) 6 yI 9. 4(3) ∀x∈Tλ∈ΛBλ |∃λ0∈Λ,s.t.x∈Bλ0 y x∈Aλ0 agx∈Tλ∈ΛAλ. XH8aj#Æ QÆ 4(4) ∀x∈Sλ∈Λ(Aλ∪Bλ) S{zλ0∈Λ,x x∈Aλ0∪Bλ0.fx∈Aλ0 agx∈(Sλ∈ΛAλ)∪(Sλ∈ΛBλ) fx∈Bλ0 aUgx∈(Sλ∈ΛAλ)∪(Sλ∈ΛBλ)..V! ∀x∈(Sλ∈ΛAλ)∪(Sλ∈ΛBλ) |x∈Sλ∈ΛAλXx∈Sλ∈ΛBλ,\nx∈Sλ∈ΛAλ,|{zCλ0∈Λxx∈Aλ0 y x∈Aλ0∪Bλ0 :x∈Sλ∈Λ(Aλ∪Bλ). j A Æ 6 y V! ∀x∈Tλ∈ΛAλC _x∈Tλ∈ΛAλ S{zCλ0∈Λxx∈Aλ0 _x∈ACλ0 y x∈Sλ∈ΛACλ08aj#Æ.V! ∀x∈Sλ∈ΛACλ0 {zCλ0∈Λxx∈ACλ0 _x∈Aλ0 a5x∈Tλ∈ΛAλ _gx∈Tλ∈ΛAλC. j Æ 9. ℄+{An} Æjpt liminfn→∞An=∞Sn=1∞Tm=nAm=∞Sn=1An _wlimsupn→∞An=liminfn→∞An=∞Sn=1An Ylimn→∞An=∞Sn=1An. ℄+{An} Æ6rt limsupn→∞An=∞Tn=1∞Sm=nAm=∞Tn=1An _wliminfn→∞An=limsupn→∞An=∞Tn=1An Ylimn→∞An=∞Tn=1An.1-1-4 (A−B)SB=(ASB)−B mT o|B=∅. m$J8a 9 STJ d gB6=∅ | /x∈B y x∈(ASB)−BWx∈(A−B)SB. l n6 Æ QÆ1-1-5nS={1,2,3,4},A={{1,2},{3,4}} [F(A) ifBS={1n|n=1,2,···},A0={{1n|nisodd}},A1={{1},{13},···,{12i+1},···} ,F(A0)IF(A1)|u7 V~℄K^PY 3Solution.F(A)={∅,{1,2},{3,4},S}F(A0)={∅,{1n|nisodd},{1n|niseven},S}F(A1)={E|E⊂S}.1-1-6 jS ℄A, ℄A F (ϕA(x)=(1,ifx∈A0,ifx∈A. fBA1,A2,···,An,···|S ℄ N+ |ϕliminfn→∞An(x)=liminfn→∞ϕAn(x),ϕlimsupn→∞An(x)=limsupn→∞ϕAn(x).Proof.1). x∈liminfn→∞Ant $U j1 wtf6\: ℄ ∃N, nNt x∈An,_ ϕAn(x)=1. YhUU j1. x∈liminfn→∞Ant $U j0 wtf6\: ℄ ∀N, {z∃nN x∈An,_ ϕAn(x)=0. YhUU j0.2). / yÆ1-1-7nf(x)| ℄zEj vF a(Vf E[fa]=∞Sn=1E[f≥a+1n],E[f≥a]=∞Tn=1E[fa−1n].Proof.7 VyÆV! L8a∀n,E[fa]⊃E[f≥a+1n] _wE[fa]⊃∞Sn=1E[f≥a+1n]..V! ∀x∈E[fa],_x∈E,f(x)a F7 nm$|t a+1nf(x) U | nm$|tx∈E[fa+1n] Yx∈∞Sn=1E[f≥a+1n] :E[fa]⊂∞Sn=1E[f≥a+1n]. Vy Æy yÆV! L8aE[f≥a]⊂E[fa−1n],∀n _wE[f≥a]⊂∞Tn=1E[fa−1n]..V! ∀x∈∞Tn=1E[fa−1n],_∀n,x∈E[fa−1n] F7x∈EW∀n,a−1nf(x) fn \Jff(x)≥a, Yx∈E[f≥a] :E[f≥a]⊃∞Sn=1E[fa−1n]. y Æ V~℄K^PY 41-1-8fBvF N+{fn(x)}zEj~’jf(x) | j\f a gE[f≤a]=∞Tk=1liminfnE[fn≤a+1k]=∞Tk=1liminfnE[fna+1k].Proof.V! ∀x∈E[f≤a],fjfn(x)→f(x)(n→∞), Y∀k,∃N, nNt |fn(x)−f(x)|1k_fn(x)≤f(x)+1k, Yfn(x)≤a+1k._w nNtx∈E[fn≤a+1k], E[f≤a]⊂∞Tk=1liminfnE[fn≤a+1k]..V! ∀x∈∞Tk=1liminfnE[fn≤a+1k], Y∀k,x∈liminfnE[fn≤a+1k],_∃N, nNt x∈E[fn≤a+1k], Y nNt x∈EWfn(x)≤a+1k,fjfn(x)→f(x)(n→∞), Yf(x)≤a+1k fk \J f(x)≤a. Yx∈E[f≤a].:∞Tk=1liminfnE[fn≤a+1k]⊂E[f≤a]. j V/ yj#Æ / y Ƨ1.2‘X\ 1-2-1dz1y1q(−1,1)I(−∞,+∞) i V/1-1 b Solution.y=tanπ2x,P x∈(−1,1).1-2-2 Tab g(a,b)∼(0,1).Proof. /y=a+(b−a)x P x∈(−1,1).| (a,b)l(0,1)p#1-1 bÆ:(a,b)∼(0,1).1-2-3 Nj J\~q qj %j ℄ |I /Nj %j ℄ Nj J# ℄I /Nj %j ℄ ÆProof.7 \~q +qIR2 Æ{vj f3 1-2-1 +q 8V lNj C ; Æy \\~q qI\~q +qC0 ÆnC( Fz(x0,y0) V~℄K^PY 5J (r \~q qÆ%mφ:C→C0,(x,y)→1r(x−x0,y−y0).8aφ(yC C0 VVm _wC∼C0. Yf 4 u J V/y4j#ÆnE( ℄Æf ℄ ℄ {zV\~q qC⊂E Yc=C≤E≤R2=c YfBernstain EI /Nj %j ℄ Ƨ1.3r ‘X1-3-1 Nj&X(g 7jV ℄KÆ nNj&X(g 7j ℄(A.g ℄Q={r1,···,rn,···}.[An={(rn,r)|r∈Q},n=1,2,···|A=∞Sn=1An.8aAn _wA 1-3-2Y ;j Q\s \i(o \℄K Dg /o z8V \i ℄Vg f n ;j Cg Q^ Cg j ℄(A |_ g ℄B ℄ fjB( ℄ YA U( ℄ daAl \i1-1 b j| QÆ1-3-3 g4 (g ?yjV ℄K \n4 ( ?y(nv ?ypn(x)=a0xn+a1xn−1+···+an−1x+an 8a(n+1/g 4 (a0,a1,···,an) _ 4 (g nv ?y _ ℄Pn={pn(x)}ln+1) iRn+1 &X(g ℄KAn={(a0,a1,···,an)} gJ An| ℄ FBPn Æ9 An dG gn=0 |An={a0} a( ℄ gnn−1tAn−1( ℄ \nAn