第一章-预备知识

1zzzzzzzzzzz2,Rab∈||||||abab+≤+||||||||abab−≥−Holder1,1pq111pq+=pq,pq,1ab11pqababpq≤+1pyx−=1qxy−=110011abpqpqabxdxydyabpq−−≤+=+∫∫Holder,pq,(1,2,,)kkxykn=R∈1/1/111||||||pqnnnpqkkkkkkkxyxy===⎛⎞⎛⎞≤⎜⎟⎜⎟⎝⎠⎝⎠∑∑∑1||0npkkx==∑1||0npkky==∑1||0npkkx=≠∑1||0npkky=≠∑1/1/||pnpkkkkaxx=⎛⎞=⎜⎟⎝⎠∑1/1/||qnqkkkkbyy=⎛⎞=⎜⎟⎝⎠∑311||||(||)(||)pqkkkknnpqkkkkxyabpxqy==≤+∑∑11111||||1(||)(||)nnpqkknkkkknnpqkkkkkxyabpxqy=====≤+=∑∑∑∑∑Minkowski1p,R,(1,2,,)kkxykn∈=1/1/1/111||||||pppnnnpppkkkkkkkxyxy===⎛⎞⎛⎞⎛⎞+≤+⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎝⎠∑∑∑qp111111/1/(1)111/1/(1)11||||||||||||||||||nnnpppkkkkkkkkkkkpqnnppqkkkkkpqnnppqkkkkkxyxxyyxyxxyyxy−−===−==−==+≤+++⎛⎞⎛⎞≤++⎜⎟⎜⎟⎝⎠⎝⎠⎛⎞⎛⎞+⎜⎟⎜⎟⎝⎠⎝⎠∑∑∑∑∑∑∑1/1/1/1111||||||||ppqnnnnppppkkkkkkkkkkxyxyxy====⎡⎤⎛⎞⎛⎞⎛⎞+≤++⎢⎥⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎝⎠⎢⎥⎣⎦∑∑∑∑1/1/1/111||||||pppnnnpppkkkkkkkxyxy===⎛⎞⎛⎞⎛⎞+≤+⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎝⎠∑∑∑HolderMinkowski[,],[,),(,]Iabab=+∞−∞(,)−∞∞(),()xtytI1/1/(1)|()()||()||()|pqpqIIIxtytdtxtdtytdt⎛⎞⎛⎞≤⎜⎟⎜⎟⎝⎠⎝⎠∫∫∫()()xtytI,pq1/1/1/(2)|()()||()||()|ppppppIIIxtytdtxtdtytdt⎛⎞⎛⎞⎛⎞±≤+⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎝⎠∫∫∫42RQZNN+1RE⊂EE,||E212{:10}nxRxx+∈+−=,∅(,)()abbaQRZRE⊂E12{,,,,}nEaaa=ERE⊂E⇔11:NEφ→EN11EZ2ZZ+N(,)()ababR21)2)3)4)Q5)1),2)3){}12,,nnnErr=,1,2,3n=51nnE∞=∪1|nmQmZn∞=⎧⎫=∈⎨⎬⎩⎭∪Q5.(,)abab{}12(,),,,,nabrrr=ε111(,)(,)22nnnnnrrabεε∞++=−+⊇∪[]12nnbaεε∞==≥−∑ε0ba−=(,)ab1.Cauchy/1{}nnx∞=0εNNε∈,mnNε||mnxxε−1{}nnx∞=CauchyCauchyRRCauchyCauchyCauchy2.RE⊂,xE∈xα≤αEλEλEsupE.EinfER()6RQRa0r(,)ararQ−+≠∅∩QRQ71902LebesgueRiemannLebesgueRiemannLebesgue1.BorelRExE∈0r(,)xrxrE−+⊂ERRREcERE=−EER⊂⇔EEBorelBorel2.ER{}nnI∈ΛRInnE∈Λ⊆∪{}nnI∈ΛE3.ER0εE{}nnI∈Λ8||nnIε∈Λ∑E(,)Iab=||IIba−4.BorelBorelBorelBorelBorelB5.Lebesgue∈EB,∈Λ∈Λ=∑()inf{||,{}}nnnnmEIIE()mEELebesgueBorelLebesgueER12,,,,nµµµE(i)1n≥1nnµµ+1,nIµ+∈'nIµ∈'II⊆(ii)nnIGIµ∈=∪1nnGE∞==∩()lim||nnImEIµ→∞∈=∑()mEELebesgue(1)QLebesgue(2)[,]ab[,)ab(,]abLebesgueba−6.Lebesgue1E()0mE≥()0m∅=2((,))([,))((,])([,])mabmabmabmab===Lebesgue3()E,FEF⊆(E)(F)mm≤4()11E,E,911(E)(E)nnnnmm∞∞===∑∪(R)m=+∞fRRa∈1((,)){R:()}faxfxa−−∞=∈ff,abR∈()1[,)fab−()1(,]fab−LebesgueE(E)m+∞,()fxE,cdEx∈()cfxd≤≤[,]cd01ncyyyd==,11([,))iiiEfyy−−=1[,),1,2,,iiiyyinη−∈=1()niiimEη=∑1max()iiiyyλ−=−01lim()niiimEλη→=∑()fxE()fxELebesgueE()()Lfxdx∫()mE=∞()fxELebesgue()fxLebesgueRiemann1,()0,ExExxEχ∈⎧=⎨∉⎩12,,,,nEEEE1()()iiifxcxχ∞==∑1()()()iiiELfxdxcmE∞==∑∫1iiEE∞==∪10Lebesgue1RiemannLebesgue2RiemannLebesgue3fE⇔||fERiemann4|()|0Efxdx=∫⇒00,()0EEmE⊆=0()0,-fxxEE=∈fEa.e.5{()}nfxEi()Fxn|()|()nfxFx≤a.e.E()FxEiilim()(),..nnfxfxae→∞=E()fxElim()lim()()nnnnEEEfxdxfxdxfxdx→∞→∞==∫∫∫lim()(),..nnfxfxae→∞=E|()|Efxdx+∞∫lim()lim()()nnnnEEEfxdxfxdxfxdx→∞→∞==∫∫∫6Fatou{}nfElim()()nnfxfx→∞=a.e.E()fxE()lim()nEEnfxdxfxdx→∞≤∫∫7{()}nfxE1()()nnfxfx+∞==∑11a.e.E|()|Efxdx+∞∫()()()nnnnEEEfxdxfxdxfxdx∞∞⎧⎫==⎨⎬⎩⎭∑∑∫∫∫8Fubini-TonelliAB(,)fxyAB×AB|(,)|fxydx×+∞∫ABABBA(,)(,)(,)fxydxdxfxydydyfxydx×==∫∫∫∫∫LebesgueRiemannLebesgueRiemann1()DxDirichlet,()Dx[0,1][0,1]()0Dxdx=∫210sinlim0nnxdxnx→∞=∫3|()|fxdx+∞−∞+∞∫|()|,nxfxdxn+∞−∞+∞∫()ˆ()()()nnnixfjxfxedxωω+∞−−∞=−∫1LebesgueRiemannLebesgue2a.e.3Lebegue121.()ijaA=N{}{}minmaxmaxminijijjjiiaa≥2.,0ab≥[0,1]θ∈,1(1)ababθθθθ−≤+−33,()1,cxxQfxxQ⎧∈=⎨∈⎩10()fxdx∫4.221sinlim1nnxxdxnx∞→∞+∫52311(1),011nnxxxxxx=−+−++−++111ln21234=−+−+6Cantor1[0,1][0,1]2(2)GGCantorG()0mG=7.ABmn×Cauchy-Shwarz2|det()|det()det()HHHABAABB≤