1、A4D7DIBAASBCA8ASBBC1A9CDASASBCA0ADBOA8ARA5BYC4C5 C0BZC21.AIB4B9LebesgueB0B1ALAWDFALBWBEB6AUA0AOB4B9 DarbouxB1CKAWC7CWA4BFAKCNA3C1 AOB4AW Darboux B1CKB4A4A2DH f (x) A4 E DFAWB7C8CEAGBOA8A0B4 E AWDCBQCEAGBEBUD : E1, E2, , En,AU max1inmEi 0DJA0S (D, f ) integraldisplayEf (x)dx, S (D, f ) integraldispla
2、yE f (x)dx.ANC7CWAFAYB1AKCNA1CMDDCCAH0, 1D9C4DFAYCLCFCIBOA8f (x) =1, xA40,1BUAWB7CKA8,0, xA40,1BUAWAJCKA8.B4CYBJBLBKA8n,BX0,1D9C4AWBEBUDn = Eni ,D3BUEni =bracketleftBigi1n ,inparenrightBig, i = 1, 2, , n 1, Enn =bracketleftBign1n , 1bracketrightBig.BFmax1inmEni = 1n 0(n ).ATS (D, f ) =nsummationdisp
3、layi=1supxEnimEni =nsummationdisplayi=11 1n = 1.CUAYBBCZA0B2 f (x)BE0,1DFCHA8BICEBWA0BLintegraldisplay0,1f (x)dx =integraldisplay0,1f (x)dx = 0.B2ANAOB4AWDarbouxB1CKAFAKCNA12. DHBECantorBX P0 DFB1B1BOA8 f (x) = 0,B6BE P0 AWBABXBUAJAH13n AWBKAKD9C4DFB1B1AHn(n = 1, 2,),A5BM f (x)CEBWBEA0D8AMBWBEBPA1C6
4、C3 f (x)A4BDBGCEAGBOA8A0B2B6BWBEDBB1A0BQAWBMD0BWBEB7ANBZCEA1DHEn A4P0AWBABXBUAJAH13n AWBKAKD9C4BOA0BFmEn =2n13n ,B2ANintegraldisplay0,1f (x)dx =summationdisplayn=1integraldisplayEnf (x)dx =summationdisplayn=1nmEn =summationdisplayn=1n 2n13n = 3.AAAZ f (x)CEBWA0D6BWBEBPAH3.3.DH f (x)BEEDFCEBWA0en = E
5、| f | n,BFlimn n men = 0.C6C3 B6B9 f (x)BEEDFCEBWA0BLAH EDFa.e.B7ANAWCEAGBOA8A0AAAZ mE| f |= = 0.CUAYBBCZA0B6en en+1, me1 mE 0,AQBE 0,AUe ED6me 0,AQBEN,A1AUn NDJA0men integraldisplayE| f (x) | dx =summationdisplayn=1integraldisplayEn| f | dx +summationdisplayn=0integraldisplayEn| f | dx summationdis
6、playn=1(n 1)mEn +summationdisplayn=0| n | mEn=summationdisplayn=1| n | mEn +summationdisplayn=0| n | mEn summationdisplayn=1mEn =summationdisplay| n | mEn summationdisplayn=1mEn,B2AHEnA4CQCQAFC6AW EAWBVBXA0summationdisplayn=1mEn = m(summationdisplayn=1En) mE 0,B6 fn BDBGCEBNmE| fn | integraldisplayE
7、|fn|fn(x)dx integraldisplayEfndx.B2ANmE| fn | 1integraldisplayEfn(x)dx,lim mE| fn | = limn 1integraldisplayEfn(x)dx = 0.BZ fn(x) 0.7.DHmE 1DJATBIBGC2A0B2ANEbracketleftBigg | fn |1+ | fn | 1 +bracketrightBigg= E| fn | ,AAAZ limn mE| fn | = limn mEbracketleftBig |fn|1+|fn| 1+bracketrightBig= 0,BZ fn 0
8、.8.DH f (x) = sin 1xx , 0 0,DH(x)A4E f AWADBJBOA8A0BFmE f integraldisplayE ff (x)dx =integraldisplayEf (x)(x)dx = 0,AAAZmE f = 0.AGAVCEBMmE f = 0,AAAZmE|f| = 0.B8B2E f nequal 0 =uniondisplayn=1EbracketleftBigg|f| 1nbracketrightBigg.AAAZmE f nequal 0 summationtextn=1mEbracketleftBig|f| 1nbracketright
9、Big= 0,BZ f (x) = 0 a.e.B9E.11.BMD0A2limnintegraldisplay(0,)dtparenleftBig1 + tnparenrightBignt 1n= 1.C6C3 AUt (0, 1)DJA0 1parenleftBig1 + tnparenrightBignt 1n 1t 1n 1t (n 2);AUt 1,)BYn 2DJA01parenleftBig1 + tnparenrightBignt 1n= 1parenleftBig1 + t + n12n t2 +parenrightBigt 1n1).5C6C3 B2AHxp1 x ln1x
10、 = (summationdisplayn=0xn)xp ln 1x =summationdisplayn=0xn+p ln 1x,B6AU x (0, 1)DJA0 xn+p ln 1x 0,BLintegraldisplay 10xp1 x ln1x dx = summationdisplayn=0integraldisplay 10xn+p ln xdx =summationdisplayn=01(n + p + 1)2 =summationdisplayn=11(n + p)2 .15.DHfnAHEDFCEBWBOA8CSA0 limn fn(x) = f (x)a.e.B9E,D6
11、integraldisplayE|fn(x)|dx 0,AQBEa , b +DFCOARBOA8(x),A1integraldisplay b+a|f (x) (x)|dx 0(AFBCDA integraldisplaye|f (x)|dx,B2ANA0limiintegraldisplayEe|fni (x)|dx = limiintegraldisplayE|fni (x)|dx limiintegraldisplaye|fni (x)|dx 0BY xn (0,),A1 limn xn = ,AT|f (xn)| 0. B2 f (x)BE(0,)DFAYBSCOARA0BFAQBE
12、 0,A1B4DCB0 x, x (0,),AU|x x| 2, i = 1, 2, .CVEni = (xni , xni+),BFEniBTAFAOC6A0B2ANA0integraldisplay(0,)|f (x)|dx integraldisplayuniontexti=1Eni|f (x)|dx =summationdisplayi=1integraldisplayEni|f (x)|dx 02summationdisplayi=1mEni =summationdisplayi=10 = ,7ANBB|f (x)|BE(0,)DFCEBWCXB5A1BL limx f (x) =
13、0.19.DH f (x)BERp DFCEBWA0g(y)BERqDFCEBWA0A5BM f (x) g(y)BERp Rq DFCEBWBEA1C6C3 f (x)BERpDFCEBWA0B2ANAAA4Rp DFCEAGBOA8A0A6ABCBBXRp Rq DFAWBOA8AXA4CEAGAWA1AGCKg(y)AXBE Rp RqDFCEAGA0B2B6 f (x)g(y)BERp Rq DFCEAGA1AU f (x), g(y)B2A4BDBGDJA0B66B1CK4, f (x)g(y)BERp RqBEDFBWBEB7B0B1A0D6integraldisplayinteg
14、raldisplayRpRqf (x)g(x)dxdy =integraldisplayRpdxintegraldisplayRqf (x)g(y)dy =integraldisplayRpf (x)dx integraldisplayRqg(y)dy 0B2AQBEA0D6AKCN integraldisplay+0F(y)dy =integraldisplayEf (x)g(x)dx.C6C3 B2 f (x), g(x)A4EDFBDBGCEAGBOA8A0AAAZB4DCB0y 0,EyA4CEAGBXA0BLF(y) = integraltextEyf (x)dxAQBE(CEAH+)D6F(y) 0.B8B6BHA9D2B1CKA0integraldisplay +0F(y)dy =integraldisplay +0integraldisplayEyf (x)dx dy = integraldisplay 0parenleftBiggintegraldisplayEEy (x) f (x)dxparenrightBiggdy =integraldisplayEf (x)parenleftBiggintegraldisplay g(x)0dyparenrightBiggdx =integraldisplayEf (x)g(x)dx.8
