Proving Positive Almost-Sure Termination
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(39) ´9³?´tBA0µ ¶^¸:,-°¸ (A, >)<¶^¸:01A0 ,-´ Ë/→) 8:3W¶^8%B93ÌBA01Aµz,-8:EFB90=Ç@Ò&¸7,´ 1A0HÁ/57,-190E µz36873FBA3=870µE=º7º7,-87· V B: =A (B, µz36873FBA3=870Dµz0mEF8:´vB9¸:EFB x → x ,Gµ`º7*-,-0H´ V (x) > V (x ) Ç L3W¶ → BA01Aµz,-8:→EFB90HB´N¹0m°E=5:´d0 EF8
(40) ,G8?Ï87,rBA0 °¸:E=,G8 a → a → ··· ¶T3657*-¿À,-8:¿?5:°0EF8
(41) ,G8?Ï87,rBA0 °¸:E=,G8 0. n. i. 1. i+1. 0. 1. 0. 0. 0. 1. V (x0 ) > V (x1 ) > . . .. Ò&¸70µ`36´dBNº3=º757*EF14°¸73=,° 0&»¼361vBA01Aµz,-8:EFB9,-3=8º71A3}3F» ´v,´NE=80µ¹0m¿7¿?,-87· ,-8/B93 >) Ç ÈPBA´^º3=º75:*-E=19,GBt³° 36µz0m´&º:EF19B9*-³»¼1A3=µ BA¸70(»¼3=*-*-3W¶^,G87·`0mE=´9³° 3=µ`º7*-0 BA0870m´9´&1A0H´957*GBÂG(N, Ä ÆSÇ MON9PRQSPUTWV4XWV(PRY[ZR\ =^]_5 2E%:,1`a8B!1') 05H*cb 2E5KJd +!&#&%I') "*1%H'),.-0/ *1%(24365 #.89#&%:,$; 305 `g8h2 ei%Eb ,$'c,d2j#k h;h305H30%:305H,,1;7!.,.-"-62E5KJl2E5 %:3 (N, >) m. %:,$'&; 2E5U 6%G,$#S2 ef 05U-. Ü ´,-8Â-ÄÆP<H36¹:´d0H19.60@B9¸:EFBB9¸:0B90m°¸787,Á65:0N,´´93=578:¿(,-8(B9¸:0N·=0H8701EF*6°E=´90=<¹75?B@°3=µ`º7*-0 BA0 3=87*-³¤»¼3=1+Ï:87,GB90H*G³Ì¹71EF8°¸7,G8:·À´9³}´dB90Hµ´ÇÈ8:¿?0H0H¿ <KBA¸70z´9³?´tBA0µ¶^,GB9¸ A = N EF8¿ → ,-´ ¿?0 Ï:8:0H¿¹}³ < <@»¼3=1`EF*-* <,-´ B90H19µ`,-8:EWBA,G8:·:< ¶^¸701A0HE6´NBA¸70(i1A0+,´&1,873`j)µz→36873F(i,BA3=k)870(0H(i,µ ¹j0H+¿7¿?1),-87·`→»¼1A3=(i,µ j)(N , →) B93 i,(N,j, k, >) Â-Ä ÆPÇ n 4¨}ª^R o
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(45) ®9@§ 0` JK,W!1') Ã3=8EÙ´d0B ,´"E´90 B"3=» { 0 BÀ5:´Ï:1´dBÀ° 36µz0¹:E=°ËÙB93Ù´A°¸73}3=* 7< }<-WÄ bÆPÛTE ´d57¹´d0BA´`3F» Ω ¶^¸7,°¸Ø°3=8/BAE=,G8:´BA¸70¤0µ`º?Bt³/ÑP´90 Bm<vEFσ8:| ¿Ã,´z´dBAE=¹7*-0
(46) ¹}³Ù°3=57Ω8/BEF¹7*-0À5:87,G368 EF8:¿°3=µ`º7*-0µ`08/BEWB9,-3=8Ç6È8Àº:EF19B9,° 57*EF1m<FBA¸70+´90 B&3F»´957¹:´90 B´N,´4E8:EFB9571EF* σÑPE=*G·60¹71E(»¼361 EF8}³À°3=578/BAE=¹7*-0 ´d0BHÇ Ü ;7,. 0#./ 'c "!1`},#I~ +*c, (Ω, σ) ,-´+E´90 BD¶^,GB9¸ E σÑEF*-·=0H¹71E368Ì,rBmÇ Ü ~U')3+!c +!$2E`g2E%(8,-´vE+»¼578°B9,-3=8 »¼1936µ E ÑEF*-·=0¹:1AEDB93 <W¶^¸7,°¸`,-´°3=578/BEF¹7*-³ E=¿:¿?,rBA,G.60=< EF8:¿
(47) ´d5°¸ÀBA¸:EWB P (Ω) = 1PÇ Ü B91A,-º7*G0σB (Ω, σ, P ) ,-´^[0,°HEF1]*-*G0mz¿ i~U')3+!. "!12E`a2E%(8#(~ "*.,Ç ÈP» (Ω, σ) EF8:¿ (Ω , σ ) EF1A0vµ`0HE6´d5:1AE=¹7*G0T´dº:E6° 0m´<bE&»¼578:° B9,-3=8 f : Ω → Ω ,f´ ;7,. 9#&/ ' | "!1`},Z,r»»¼3=1TE=*G* ,-8 σ < f (W ) ∈ σ Ç Ü ') 65U-"36;D0 0'&24 "!1`},,-´TEµ`0HE6´d571EF¹:*G0^»¼5:8:°BA,G368 W 2. 2. 0. 0. 0. 0. −1.
(48) 3=8Ô´93=µ`0zº71A3=¹:E=¹7,G*-,GBt³¤´9º:E6° 0=ÇKÒ&¸7F0 ;7,. 65 3F»TEÀ1AE=8:¿?36µ .WE=19,EF¹7*-0 BAE=Ë},G87·
(49) .WEF*-570H´Z,-8 9B ¸70(´d0B N 3F»,-8/B90·601´v,´ E[V ] = P iP (V = i). Ò&¸7,-´4.WEF*-570Z,-´4EF*-V¶&Eb³}´T¿?0 Ï870H¿ <?0H.=0H8 ,r»,GB^°EF8"¹0+Ï:87,GB90Z3=14,G87Ï:87,GB90=Ç:)+¹:´901A.=0B9¸:EFB&´95:°¸"E 1AE=8:¿?3=µ .WEF1A,-E=¹7*-0+EF*-¶&Eb³}´N´9EFB9,´t»¼³ B9¸70´93FÑ°E=*G*-0H¿^%G,1`},$#$*.3.~,fe$30'&; / `} E[X] = P P (X > n)  ÕFÆSÇ:>:3=1Ez1EF8:¿?36µß.WEF1A,rÑ EF¹7*-0 BAEFË},-87·.WEF*-570m´,-8 <7BA¸70 µ`0mEF8 E[V ] °E=8 ´tBA,G*-*@E=*G¶&Eb³?´^¹0z¿?0Ï:870m¿ Û º71AE6°BAV,-°HEF*-*G³6<7,rBZ,-´D,G87Ï:87,GB90,GN» P∪(V{+∞} E=8:¿¤0HÁ/5:E=*BA3 E[V ] = P iP (V = i) ¶^¸7,°¸¤µ`Eb³À´tBA,G*-* ¹0,-8?Ï:8:,rBA0 &3=B9¸7=0H19¶^+∞) ,´d06Ç > 0 CZ,G.608 A, B ∈ σ <&¶^¸708 P (B) > 0 B< %Eb , *c365U-62E%(24365U 0`~H')3"!. "!12E`a2E%I8 3 e A J62ED0,15 , ´z¹/³ ¿?0Ï:87,GB9,-3=8 P (A|B) = P (A ∩ B)/P (B) ÇÒ&¸70
(50) µ`0HE=8Ù3F»1EF8¿?3=µÚ.WEF1A,-E=¹7*-0 B ° 3=8¿?,rBA,G36870H¿À¹}³ B ,´^¿?0 Ï:8:0H¿
(51) ¹}³ E[V |B] = P iP (V = i|B) Ç V : ÜΩ → N #&%G3*.b 9#&%I24*l#1,1/ ,15U*., 305 ^#$,1% ,´zEÌ» EFµ`,-*G³ <@3F»1EF8:¿73=µ .bE=19,EF¹:*G0m´ ¿?0 Ï:8:0H¿368Ù´93=µ`0Ï7½?0H¿º71936¹:EF¹:,G*-,rBt³ ´9Aº:E=°0 (Ω, σ, P )(X¶^,GB9)¸Ù.bE=*G5:0H´3=8 A ÇÈPBz,-´`´AEF,¿ B93%¹0 ^ 0'+30D24 05£,r»Z,rB´° 368:¿?,GB9,-3=8:E=*&¿7,-´dB91A,G¹:5?B9,-3=8û¼578:° B9,-3=8Þ´9EFB9,´tÏ:0m´B9¸70 ´93FÑ°E=*G*-0H¿ EF1AË=3W.`º71A3=º019Bt³=<}BA¸:EWB,´4»¼3=1E=*G* EF8:¿ < n s∈A i. ∞ n=0. i. i i i∈N. P (Xn = s|X0 = π0 , X1 = π1 , . . . , Xn−1 = πn−1 ) = P (Xn = s|Xn−1 = πn−1 ),. FE 8:¿ b36;73cJ+,15U,.36/ #,r»@»¼5719B9¸70H19µ`3=1A0+B9¸7,´^º71A3=¹:E=¹7,-*G,GBt³,´&,G8:¿70º08:¿?0H8/B3F» n Ç Ò&¸70¤µ`EFB91A,r½ ,´`¶^¸:EWB,´°E=*G*-0H¿ØE%´tBA3}°¸E=´dB9,° µ`EFB91A,r½ 0.608z¶^¸7(p08 A) ,-´=E=8(P,-8?(XÏ:87,GB90´9=0 B %t|X ÕWÆSÇWÈP=Bv¸s))E=´B9¸70&87,° 0&º71A3=º019Bt³(B9¸EWBv° 36*G57µ`8:´ ´d57µÊB93 1 Ç CZ,G.},-87· . E ^ 0'+30D
(52) b 02E5 ,-´T3F»°3=571´d0D0HÁ/57,-.bE=*G0H8/BNB93·=,-.},G87·B9¸70Z´d0mÁ65:08:°0D3F»,GBA´ ´tBA3}°¸E=´dB9,°µEWBA19,° 0m´Ç7CZ,-.=08
(53) E 30;h3)JK,15U,.30/ # 6'"36D
(54) b 02E5°3=1A190m´dº3=8:¿:´vBA3·6,G.},-87· E5:87,-Á/570´dB93?°¸:E6´tBA,-°ZµEWBA19,G½ EWBEF8}³1EF87ËK<}BA¸70µEWB91A,G½",-´4BA¸70´AEFµ`0 Ç ^ ª = ¨ © 4 ¨ o ªv© . Î 0E=190&´d0mEF1°¸7,-87·Z°19,GB90H19,E+,G8zB9¸:0´dº7,-1A,rB3F» 1A3=º36´9,rBA,G368ÀÄ6ÇF>7361B9¸:EFBvº7571Aº3/´d06<W¶T0&873W¶ ´tBEWB90B9¸70D»¼3=*-*G3W¶^,-87·1A0H´957*rBm<FBA¸:EWB&°EF8`¹0+EFBdB91A,-¹75?B90m¿`B93>736´dB90H1 WÆPÇ=ÈPBT¸E=´N´dB91A3=8:·° 3=87Ñ 870H° B9,-3=8:´¶^,rBA¸ E=1dBA,G8:·6EF*-0BA¸703619³(EF8:¿ °EF8 ¹0&´d0H08 E6´E+° 3=8´d0mÁ65:08:°0v3F»7.601A³+·60870H1AE=* 190m´d57*GBA´+3F» ´957º0H1 E=1dBA,G8:·6EF*-0BA¸703619³6Ç D3W¶T0H.=0H1H<,rB°E=8 ¹0`º71A3W.=0m¿¤,G8¿?0º08:¿708/B9*-³=< EF8:¿"B9¸70º:193}3F»@¸701A0(,-´DE=¿7E=º?B90m¿»¼1A3=µ  ÕFÆPÇ i s,t. i+1. i. P N R $f2ED0,15 2j#. Z PUTWX N!}T PRN #" \ b30#1, ;7 6%('&2'& 7b36;73cJ+,15U,.36/ #^ 0'+30D *.b 02E5s36D9,1' F*.30/ 5 %: "!1`},k#I~ +*., A % 2 e %>b,1'), ,&+2j&# %4# z;7,. 9#&/ ') +!$` , #&/ !,1% 65U- #130;h,de./ 5H*$%I24305 P = (pt,s ) ( 2E%>b C ⊂ A( V : A → R ()%. inf V (i) > −∞. 05U-h#&/ *cb %Eb 0%f%Eb ,d;h,. 05. i∈A. -6'&2 e.% -+,4]S5U,.- !18. ∆V (i) =. X. pi,k V (k) − V (i). k∈A. #$ 6%(2j#>]f,&#_e$30'O#136;7,. >0 ∆V (i) ≤ −. e$30' 6`E`. i 6∈ C,.
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(56) B9¸70¤873=BAEWBA,G368 »¼3=1 ÇN¯},G8° 0 <3=870 µ`Eb³E=m ´A´d57µ`0^¶^,GB9¸`*-36´A´3F»K·=08:01XEF*-,rBt³BA¸:EW(XB V, .≥. .0, X¹/³z)E=¿7¿7,G87·Einf°3=8:V´dBA(i) E=86B>,G» 87−∞ 0H°0H´A´9E=19³6Ç Ö4EF*-* τ B9¸70`1A0 B95:198B9,-µz0B93 C <¿?0 Ï870H¿ÔE=´ τ = +∞ ,r» X 6∈ C »¼361(E=*G* n < EF8¿ 3FBA¸701A¶^,´d06Ç ,-´N¶^¸:EWB4,-´T°EF*-*-0H¿ k#&%G3.~"~U2E5 J7%I2E;7,6Û/,rB4,-´TE τ1AE=8:= ¿?3=min{n µ .WE=19,EF≥¹7*-0,04B9X¸:EFB∈BAEFC}Ë60H´.WEF*-570&,-8 Nτ∪{+∞} <bE=8:¿»¼3=1EF*-* n <WBA¸70&0.=0H8/B {τ = n} °EF8
(57) ¹00 ½?º71A0H´A´d0m¿",-8"BA01Aµ´^3F» X , . . . , X Ç Î ¸708 ,´E`´90 BH<¿?0873=B90 »¼3=1^,GBA´D°¸:EF1E=° B901A,´tBA,-°»¼578:°BA,G368ÇÖT368:´d,¿?0H1^1AE=8:¿?3=µ .bE=19,EF¹:*G0 Y F= V (X )1 ÇÒ&¸:10v,G8:0HÁ/5:EF*-,GBt³,G8(BA¸70v¸}³}º3=B9¸70m´d,´ ,´}Ðt5:´dB E[V (X )|X = »¼361EF*-* Ç i] ≤ V (i) − >73=1E=8 i 6∈ C < i ∈/ C n 0. . 0. n. i. C. C. n. n. C. C. 0. n. F. n. n. n<τ. n+1. Ei [Yn+1 |X0n ] = Ei [Yn+1 1n<τ |X0n ] + Ei [Yn+1 1n≥τ |X0n ] = Ei [Yn+1 1n<τ |X0n ] ≤ Ei [V (Xn+1 )1n<τ |X0n ] = 1n<τ Ei [V (Xn+1 )|X0n ] = 1n<τ Ei [V (Xn+1 )|Xn ]. n. < B9¸70B9¸7,-1A¿É0mÁ65EF*-,rBt³°3=µ`0H´»¼1936µ B9¸70» E=°B Î ¸701A0 ´dBAEF8¿7´Z»¼3=1 B9¸:EFB 1 ,-E´E+[.]»¼578:° B9,-3=83F» XE[.|X <FE=8:¿BA=¸704i]»¼3=5719B9¸0HÁ/5:E=*G,GBt³(»¼1936µB9¸70 E=19Ë63W.+º:1936º0H1dBt³6Ç Î 036¹?BAE=,G8 0. i. n 0. n<τ. ´d,-8:° 0E=*Gµ`36´dB´d5:190H*G³ Ò&¸701A0 »¼36190. Ei [Yn+1 |X0n ] ≤ 1n<τ V (Xn ) − 1n<τ Xn 6∈ C. »¼3=1. n<τ. Ç. Ei [Yn+1 |X0n ] ≤ Yn − 1n<τ .. ÒE=Ë/,-87·z0 ½?º0H°BEWBA,G368:´. 0 ≤ Ei [Yn+1 ] ≤ Ei [Yn ] − Pi (τ > n),. ¶^¸701A0 P (.) ´tBEF8:¿:´4»¼3=1 P (.|X = i) Ç ÈPB901EWBA,G8:·^B9¸70&EF¹3W.=0N0HÁ/5:EF*-,GBt³=<bEF8:¿36¹:´901A./,-87·^BA¸:EWB Y ,-´@87368?ÑS8:0·6EFB9,-.=06< ¶T0T3=¹?BEF,-8 0. i. n. 0 ≤ Ei [Y0 ] − . n X. Pi (τ > k).. 2T57BNEF*-µz3/´tBv´9571A0*-³ ¶^¸:08 <FEF8¿ P P (τ > k) = E [τ ] »¼1A3=µ B90*-0H´A° 36º0Z»¼3619µ57*-E:Ç Y = V (i) X = i Ò&¸701A0 »¼361906</¶40(¸:Eb.=0 E [τ ] ≤ V (i)/ Ç L3FBA,-°0`B9¸:EFBB9¸70B90H°¸:87,-Á/5703=»45´d,-87· >73/´tBA01 ´ZBA¸7036190Hµ ,-8%3=1¿?01(BA3̺7193W.60° 3=87Ñ .=01A·=0H8:° 0B93´93=µ`0D´90 B C ¸:E6´T´9,Gµ`,-*-E=19,GB9,-0H´N¶^,rBA¸BA0H°¸787,Á/570H´N5:´90H¿,-8"´90*G» ÑP´dBAE=¹7,-*G,-;HEFB9,-3=8 ÂGÄWÓ/<GÄ6ÄÆPÇ k=0. 0. 0. i. ∞ k=0. i. i.
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(95) B9¸70°3=1A190m´dº3=8:¿7,G87·ÀB91EWÐt0m°BA3=1A³Ì3F» Ma 3F»^´9E=µ`0z*-08:·FB9¸ ´dBAE=1dBA,G87·
(96) »¼1A3=µ a *G0mE=¿7´ZφBA3¤E B901Aµ`,G8EF*7¸7,´dB93=1A³=Ç6>75719B9¸70H19µ`361906<WB9¸70Dº71936¹:EF¹:,G*-,rBA,G0m´3F»°3=1A190m´dº3=8:¿7,G87·(¿?0H19,-.WEWB9,-3=8´vEF1A0 º7190m´d0H19.60H¿ Ç ÖT3=8:´9,-¿701&873W¶»¼5:8:°BA,G368 W : S → R ¿70 Ï:870m¿
(97) ¹/³. ph,h = 1. φ. φ. φ. W (a0 a1 · · · an ) = V (an ). »¼3=1EF*-* 1A0HEF*-,-;HEF¹:*G0Z¸7,´tBA3=1A³ a a · · · a Ç Î 0¸:Eb.60 ∆W (h) = ∆ V (h) ≤ − »¼3=1EF8}³8:3=8?ÑSB90H19µ`,-8:EF*K1A0HEF*-,-;HEF¹:*G0(¸7,´tBA3=1A³ h <:¶^¸701A0 µ = φ(h) Ç Î 0z°EF8 B9¸70H8ÔEFº:º7*G³ÌÒ&¸70H3=1A0µ Ä 368 <¶^,GB9¸ 0HÁ/5:E=*BA3B9¸:0z´90 BZ3F»B90H19µ`,-8:EF* 190mEF*-,G;mEF¹7*-0^¸7,´tBA3=1A,G0m´B93 °3=8:°*G5¿?0^B9¸EWBvBA¸70+M¿?0H19,-.WEWBA,G368:C´v´dBAEF19B9,-87·(»¼1A3=µ a ,-8 M 190mE=°¸ B901Aµ`,G8EF*?1A0HEF*-,-;HEF¹:*G04¸7,´tBA3=1A,G0m´,-8E+B9,-µz0^¶^¸:36´90Tµ`0HE=8,´*-0H´A´@B9¸:E=8 W (a)/ = V (a)/ Ç 08° 0=</EF*-*7BA¸70+¿?0H19,-.WEWBA,G368:´v´dBAEF19B9,-87·»¼1A3=µ ,-8 A 578¿?01Tº36*G,° ³ φ 190mE=°¸zB90H19µ`,-8:EF* ´tBEWB90m´N,-8¤EB9,-µ`0(¶^¸736´90Dµ`0HE=8",´TE=*-´93 *-0H´A´TB9¸:E=a8 V (a)/ Ç}Ò&¸7,-´4¸736*-¿:´N»¼3614E=*G* a E=8:¿ φ Ç Î 0(873W¶º:193W.60B9¸EWB&B9¸70(BA0H°¸78:,-Á/570(,´4°3=µ`º7*-0 BA0+»¼361&Ï:87,GB90H*G³¹:1AE=8:°¸7,-87·z´9³?´tBA0µ´HÇ 2j#q]S5 2E%:,$`g8 Y V4XWV4PUY \ = ~U'c3"!. "!12E`a2j#&%(24*F "!$#&%('c "*1%@'),.-0/ *1%I24305 #&89#&%G,1; A = (A, →) 0 1. n. µ. φ. φ. . . !$') 65U*.b 2E5 J^2 ee$30'l 6`E`. a → µm. . %>b,1'), 2j#l 0%B;730#.%d ]_5R2E%G, 5 / ;h!.,$'3 eF-02j#&%I'&24!$/ %I24305 #. a(. .
(98) . µ %. 2E%Eb. . P Q X Y TT GP N Y V>X N Y V(Y sT TWX ! T " \ ek ]_5R2E%G,1`a8 PRN !$') 65U*.b 2E5 JO~H')3"!. "!12E`a2j#&%I24*h "!$#.%I') +*$%f'),.-0/ *1%(24365 #&89#&%G,1; 2j# @ # %:,1'&;2E5H 0% A= (A, →) m m | 2E%Eb 65U2E5KJ %>b,15%Eb ,$'),^,&+2j#&%O#130;h,Oe./ 5U*1%(24365 V : A → R ( % inf V (i) > −∞ ( #$36;7, #&/ *cb %>b 6% e$30'h 0`E` #&%: 0%:, e$30'h 0`E` 2E%>b i∈A %>b,h-0'&2 e.%2E5 > 0( a ∈ A( µ % a → µ( a ( "*.*.30')-62E5KJl%G3 -",(]_5U,.-l!$8. . µ. ∆µ V (a) =. X. µ(i)V (i) − V (a). i. #$ 6%(2j#>]f,&#. ∆µ V (a) ≤ −.. 2T³ ¸/³}º3FBA¸70H´9,-´H<b»¼3=1vE=*G*´tBEWB90 <=EF8¿ º3=*-,-°³ <=¶40&¸:Eb.=0 Ç Î ¸708 ,h ´E¤m 1A0HE=*G,-;HE=¹7*-0¸7,´dB93=1A³=<E=8:¿ φ ,´aE
(99) º36*G,° ³6<¶T0`¶^φ 1A,rBA0 T [h, φ]T [a,»¼361(φ]B9¸7<0+∞ µ`0HE=8ÉB9,-µz0 ¹0»¼3=1A0Z1A0HE6°¸7,-87· ⊥ EW» BA01¸7,´dB93=1A³ h Ç L3FBA0 BA¸:EWB(»¼361(E=8}³
(100) º3=*-,° ³ < ¶^¸70H8 h ,´ZEÀ1A0HEF*-,-;HEF¹:*G0873=8?ÑSB90H19µ`,-8:EF*¸7,´dB93=1A³=<¶40 φ ¸:Eb.=0 X Ä T [h, φ] = 1 + φ(h)(x)T [hx, φ] . ')33 e. x∈A.
(101) ÈP»@º3=*-,-°³ φ ,´ =E 19Ë63W.},-E=8<6¶40(¸:Eb.=0. <7EF8:¿
(102) ¸70H8:° 0. T [hx, φ] = T [x, φ] X φ(h)(x)T [x, φ]. T [h, φ] = 1 +. Í Ò&¸70(,-¿?0mEz,-´4BA3° 3=8´d,¿?014BA¸70 t¶43=1´dB +´dB91EWB90H·=³ Ç:Ò&¸:,-´´dB91EWBA0·6³°EF8
(103) ¹0¹757,-*rBDE6´ »¼3=*-*G3W¶´HÛ,-8ÙEF8}³1A0HE=*G,-;HE=¹7*-0873=8?ÑSB90H19µ`,-8:EF*N¸7,-´dB93619³ Φh = a . . . a < Φ µEFº´ h BA3ÌB9¸:0 ¿?,-´dB91A,-¹75?B9,-3=8 µ ¶^,rBA¸ a → µ B9¸:EFBµ`EF½?,Gµ`,G;H0H´ sup P µ(x)T [hx, φ] Ç ¯},G8° 0ZB93E=8/³À´tBA1AEFB90·6³ φ 368
(104) °HEF8¤E=´A´d3?° ,EWBA0ZE´dB91EWBA0·6³ φ ¶^,rBA¸ x∈A. 0. n. BAE=Ë=0. φ. x∈A. n. 0. T [hx, φ] = T [x, φ0 ]. φ0 (h0 ) = φ(hh0 ) sup. »¼3=1E=8}³190mEF*-,G;mEF¹7*-0Z873=8?ÑSB90H19µ`,-8:EF* ¸:,-´dB93619³ h < 0. X. µ(x)T [hx, φ] = sup. X. µ(x)T [x, φ],. EF8:¿À¸708° 0 Φ ,-´ E=19Ë63W.},-E=8Ç Î 0°*-E=,Gµ BA¸:EWB^BA¸7,´^,-´&,-8:¿?0H0H¿ÀB9¸:0¶T361A´dB&´dB91EWBA0·6³=<}, Ç 0=Ç φ x∈A. φ x∈A. ±. sup T [h, φ] ≤ T [h, Φ]. »¼3=1EF*-* 1A0HEF*-,-;HEF¹:*G0Z873=87Ñ BA01Aµz,-8:E=* ¸7,-´dB93619³ h Ç Ò&¸7,-´v»¼3=*-*G3W¶´»¼1936µ BA¸70^»¼3=*-*-3W¶^,G87·E=19·657µ`08/BA´HÛW»¼361TEF8}³,G8/B90H·=0H1 <=*-0 B ¹0B9¸70Z´d0B 3F»^´tBA1AEFB90·6,G0m´+B9¸:EFB° 36,G8:°,-¿70H´(¶^,GB9¸ Φ 3=8%EF*-*v¸7,´tBA3=1A,G0m´(3F»&*G0H87·FBA¸Éi*-0H´A´ZB9Φ¸EF8 i ÇD´9,G8:· 190Hº0mEWB90m¿?*-³ NÁ/5:EWBA,G368%Ä=<?3=8:0Z·60 B´4»¼3=1EF*-* ,-86BA0·601 i < φ. i. sup T [h, φ] ≤ sup T [h, φ]. »¼3=1EF*-* 1A0HEF*-,-;HEF¹:*G0Z873=87Ñ BA01Aµz,-8:E=* ¸7,-´dB93619,-0H´ h 3=»@*-08:·FB9¸
(105) *-0H´A´4B9¸:E=8 i Ç L3W¶<T´9,-8:° 0 ,´BA¸70*-,Gµ`,GB
(106) 3=» sup T [h, φ] ¶^¸708 i 6· 3/0m´BA3Ù,-8?Ï:87,GBt³=< NÁ/5:EWBA,G368¤±z¸736*-¿:T´Ç[h, Φ] L3W¶<?,-8¤E=8}³873=8?ÑSB90H19µ`,-8:EF* <7¶^,GB9¸ < a a→µ P P µ(x)T [x, Φ] ≤ sup P µ(x)T [x, φ] Õ ! ≤ sup Φ(a)(x)T [x, φ] P ≤ Φ(a)(x)T [x, Φ]. ¶^¸701A0Ï:1´tB,G870mÁ/5:EF*-,rBt³É° 36µz0m´»¼1A3=µ B9¸70» E=°BB9¸:EFB Φ ,´ E̺:E=1dBA,-°57*EF1 ´dB91EWBA0·=³6< B9¸:0 ´d0m° 3=8¿
(107) »¼1936µßB9¸:0¿?0Ï:87,GB9,-3=8Ì3F» <EF8:¿
(108) BA¸70B9¸:,G1¿
(109) »¼1936µÊBA¸70» E6°BBA¸:EWBDB9¸70z´d57º 3F»Ez´957µß,´E=*G¶&Eb³?´N*-0H´A´&B9¸EWB&B9¸7Φ(a)(x) 0´957µ3F»@B9¸70´957º:´HÇ »¼3=1E=*G*´dBAEFB90 <:EF8:¿ <?¶40 Î 0E=190(¿?3=8:0=Û7,-8:¿?0H0H¿ <7,G»¶T0ZBAE=Ë=0 Ë/8:3W¶B9¸EWB V ,-´873=8?ÑP870H·6EWBA,G.60=<}EF8:¿À»¼36V1(a)EF8}³ =µT¶^[a,,GB9¸ Φ]a → µ <7¶T0(¸:Eb.6a0 = 1 φ. φ∈Φi. φ∈Φi. x∈A. φ. x∈A. φ. x∈A. x∈A. P ∆µ V (a) = Px∈A µ(x)V (x) − V (a) [x, Φ] − T [a, Φ]P = x∈A µ(x)T P = −1 + ( x∈A µ(x)T [x, Φ] − x∈A Φ(a)(x)T [x, Φ]) ≤ −1. ¶^¸701A0ZB9¸7,-1A¿À0HÁ/5:E=*G,GBt³° 36µ`0H´4»¼1A3=µ NÁ/5:EFB9,-3=8ÌÍ?<7E=8:¿"*E=´dB^,G8:0HÁ/5:EF*-,GBt³`»¼1936µ NÁ/5:EWBA,G368 Õ:Ç.
(110) ?@,1;7 6' L3=B90NB9¸:EFBBA¸70T1A0H´dB91A,°B9,-3=8(BA3Ï:8:,rBA0*-³¹71AE=8:°¸7,-87·D´9³?´tBA0µ´,-8BA¸70Tº:190H./,-3=5´ B9¸70H3=1A0µÊ,´m µE=8:¿7EWBA3=1A³KÛ?B9¸7,´D°HEF8̹0 ´9008 E6´E°3=8:´90HÁ/570H8:° 03=» 0µEF1AËÀ±:Ç:È8:¿?0H0H¿ < ° 3=8´d,¿?01+B9¸:0`°3=578/BA019ÑS0½?E=µ`º7*G0EW» B90H1 1936º3/´d,GB9,-3=8 1 < ° 368:´d,¿?0H190m¿ E=´ZE"º:1936¹:EF¹7,-*-,-´dB9,° EF¹:´dB91E=° B190m¿?5:°BA,G368Ì´d³?´dB90µ¤Ç7ÈP»BA¸701A0¶401A0(Ez»¼5:8:°BA,G368 EF8:¿Ì´93=µ`0 E=´^,-8¤B9¸70 ° 3=8° *-5:´d,-3=8À3F»º71A0.},G365:´TB9¸70H3=1A0µ¤<?E=¿7¿7,G87·E`° 368:´dBAEF8/B&,r»V870m° 0H´A´AEF1A³=<?EF8:¿À>µ 05:*rBA,Gº7*-³},G8:· ¹/³ 1/ ,G»Z8:0H° 0m´9´AEF1A³=<¶40
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