Copyright (c) 1992 Association of Mizar Users
environ vocabulary SUPINF_1, MEASURE1, RLVECT_1, ARYTM_3, ARYTM_1, SETFAM_1, FUNCT_1, BOOLE, TARSKI, RELAT_1, SUPINF_2, PROB_1, MEASURE3, MEASURE4; notation TARSKI, XBOOLE_0, ENUMSET1, ZFMISC_1, SUBSET_1, NUMBERS, XREAL_0, RELAT_1, FUNCT_1, FUNCT_2, REAL_1, NAT_1, SETFAM_1, SUPINF_1, SUPINF_2, MEASURE1, MEASURE3; constructors ENUMSET1, NAT_1, REAL_1, SUPINF_2, PROB_2, MEASURE3, MEMBERED, XBOOLE_0; clusters SUBSET_1, FUNCT_1, SUPINF_1, MEASURE1, RELSET_1, NAT_1, MEMBERED, ZFMISC_1, XBOOLE_0, NUMBERS, ORDINAL2; requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM; definitions TARSKI, MEASURE3, XBOOLE_0; theorems TARSKI, ENUMSET1, FUNCT_1, FUNCT_2, ZFMISC_1, NAT_1, SUPINF_1, SUPINF_2, MEASURE1, MEASURE2, MEASURE3, RELSET_1, SETFAM_1, XBOOLE_0, XBOOLE_1, XCMPLX_1; schemes NAT_1, FUNCT_2, XBOOLE_0; begin :: Some theorems about R_eal numbers reserve x,y,z for R_eal, A,B,X for set, S for sigma_Field_Subset of X; theorem Th1: 0.<=' x & 0.<=' y & 0.<=' z implies (x + y) + z = x + (y + z) proof assume A1: 0.<=' x & 0.<=' y & 0.<=' z; A2:(x in REAL or x in {-infty,+infty}) & (y in REAL or y in {-infty,+infty}) & (z in REAL or z in {-infty,+infty}) by SUPINF_1:def 6,XBOOLE_0:def 2; per cases by A1,A2,SUPINF_2:19,TARSKI:def 2; suppose A3:x is Real & y is Real & z is Real; x + y is Real & y + z is Real proof not x = -infty & not x = +infty & not y = -infty & not y = +infty & not z = -infty & not z = +infty by A3,SUPINF_1:31; then not x + y = -infty & not x + y = +infty & not y + z = -infty & not y + z = +infty by SUPINF_2:8,9; hence thesis by MEASURE3:2; end; then consider a,b,c,d,e being Real such that A4:x = a & y = b & z = c & x + y = d & y + z = e by A3; (x + y) + z = d + c by A4,SUPINF_2:1 .= (a + b) + c by A4,SUPINF_2:1 .= a + (b + c) by XCMPLX_1:1 .= a + e by A4,SUPINF_2:1 .= x + (y + z) by A4,SUPINF_2:1; hence thesis; suppose A5:x = +infty; A6:not (y + z) = -infty by A1,SUPINF_2:9,19; (x + y) + z = +infty + z by A1,A5,SUPINF_2:19,def 2 .= +infty by A1,SUPINF_2:19,def 2 .= +infty + (y + z) by A6,SUPINF_2:def 2; hence thesis by A5; suppose A7:y = +infty; then (x + y) + z = +infty + z by A1,SUPINF_2:19,def 2 .= +infty by A1,SUPINF_2:19,def 2 .= x + (+infty) by A1,SUPINF_2:19,def 2 .= x + (+infty + z) by A1,SUPINF_2:19,def 2; hence thesis by A7; suppose A8:z = +infty; not (x + y) = -infty by A1,SUPINF_2:9,19; then (x + y) + z = +infty by A8,SUPINF_2:def 2 .= x + (+infty) by A1,SUPINF_2:19,def 2 .= x + (y + (+infty)) by A1,SUPINF_2:19,def 2; hence thesis by A8; end; theorem Th2: (not x = -infty & not x = +infty) implies (y + x <=' z iff y <=' z - x) proof assume A1:not x = -infty & not x = +infty; A2:(x in REAL or x in {-infty,+infty}) & (y in REAL or y in {-infty,+infty}) & (z in REAL or z in {-infty,+infty}) by SUPINF_1:def 6,XBOOLE_0:def 2; hereby assume A3: y + x <=' z; (y + x) - x = y proof per cases by A2,TARSKI:def 2; suppose y in REAL; then consider a,b being Real such that A4:a = x & b = y by A1,A2,TARSKI:def 2; y + x = b + a by A4,SUPINF_2:1; then consider c being Real such that A5:c = y + x; (y + x) - x = c - a by A4,A5,SUPINF_2:5 .= (b + a) - a by A4,A5,SUPINF_2:1 .= b + (a - a) by XCMPLX_1:29 .= b + 0 by XCMPLX_1:14 .= y by A4; hence thesis; suppose A6:y = -infty or y = +infty; now per cases by A6; suppose A7: y = -infty; hence (y + x) - x = -infty - x by A1,SUPINF_2:def 2 .= y by A1,A7,SUPINF_2:7; suppose A8: y = +infty; hence (y + x) - x = +infty - x by A1,SUPINF_2:def 2 .= y by A1,A8,SUPINF_2:6; end; hence thesis; end; hence y <=' z - x by A1,A3,SUPINF_2:15; end; assume A9:y <=' z - x; (z - x) + x = z proof per cases by A2,TARSKI:def 2; suppose z in REAL; then consider a,b being Real such that A10:a = x & b = z by A1,A2,TARSKI:def 2; z - x = b - a by A10,SUPINF_2:5; then consider c being Real such that A11:c = z - x; thus (z - x) + x = c + a by A10,A11,SUPINF_2:1 .= (b - a) + a by A10,A11,SUPINF_2:5 .= b - (a - a) by XCMPLX_1:37 .= b - 0 by XCMPLX_1:14 .= z by A10; suppose A12:z = -infty; hence (z - x) + x = -infty + x by A1,SUPINF_2:7 .= z by A1,A12,SUPINF_2:def 2; suppose A13:z = +infty; hence (z - x) + x = +infty + x by A1,SUPINF_2:6 .= z by A1,A13,SUPINF_2:def 2; end; hence thesis by A1,A9,SUPINF_2:14; end; theorem (0. <=' x & 0. <=' y) implies x + y = y + x; :: Some additional theorems about measures and functions theorem Th4: for S being non empty Subset-Family of X, F, G being Function of NAT,S, A being Element of S st for n being Element of NAT holds G.n = A /\ F.n holds union rng G = A /\ union rng F proof let S be non empty Subset-Family of X; let F, G be Function of NAT,S, A be Element of S; assume A1:for n being Element of NAT holds G.n = A /\ F.n; thus union rng G c= A /\ union rng F proof let r be set; assume r in union rng G; then consider E being set such that A2:r in E & E in rng G by TARSKI:def 4; consider s being set such that A3:s in dom G & E = G.s by A2,FUNCT_1:def 5; reconsider s as Element of NAT by A3,FUNCT_2:def 1; r in A /\ F.s by A1,A2,A3; then A4:r in A & r in F.s by XBOOLE_0:def 3; F.s in rng F by FUNCT_2:6; then r in union rng F by A4,TARSKI:def 4; hence thesis by A4,XBOOLE_0:def 3; end; let r be set; assume r in A /\ union rng F; then A5:r in A & r in union rng F by XBOOLE_0:def 3; then consider E being set such that A6:r in E & E in rng F by TARSKI:def 4; consider s being set such that A7:s in dom F & E = F.s by A6,FUNCT_1:def 5; reconsider s as Element of NAT by A7,FUNCT_2:def 1; A /\ E = G.s by A1,A7; then A8:r in G.s by A5,A6,XBOOLE_0:def 3; G.s in rng G by FUNCT_2:6; hence thesis by A8,TARSKI:def 4; end; theorem Th5: for S being non empty Subset-Family of X for F, G being Function of NAT,S st (G.0 = F.0 & for n being Element of NAT holds G.(n+1) = F.(n+1) \/ G.n) holds for H being Function of NAT,S st (H.0 = F.0 & for n being Element of NAT holds H.(n+1) = F.(n+1) \ G.n) holds union rng F = union rng H proof let S be non empty Subset-Family of X; let F, G be Function of NAT,S; assume A1:G.0 = F.0 & for n being Element of NAT holds G.(n+1) = F.(n+1) \/ G.n; let H be Function of NAT,S; assume A2:H.0 = F.0 & for n being Element of NAT holds H.(n+1) = F.(n+1) \ G.n; A3:for n being Element of NAT holds H.n c= F.n proof let n be Element of NAT; A4:n=0 implies H.n c= F.n by A2; (ex k being Nat st n = k + 1) implies H.n c= F.n proof given k being Nat such that A5:n = k + 1; H.n = F.n \ G.k by A2,A5; hence thesis by XBOOLE_1:36; end; hence thesis by A4,NAT_1:22; end; A6:union rng H c= union rng F proof let r be set; assume r in union rng H; then consider E being set such that A7:r in E & E in rng H by TARSKI:def 4; consider s being set such that A8:s in dom H & E = H.s by A7,FUNCT_1:def 5; reconsider s as Element of NAT by A8,FUNCT_2:def 1; A9: E c= F.s by A3,A8; F.s in rng F by FUNCT_2:6; hence thesis by A7,A9,TARSKI:def 4; end; thus union rng F c= union rng H proof let r be set; assume r in union rng F; then consider E being set such that A10:r in E & E in rng F by TARSKI:def 4; consider s being set such that A11:s in dom F & E = F.s by A10,FUNCT_1:def 5; reconsider s as Element of NAT by A11,FUNCT_2:def 1; ex s1 being Element of NAT st r in H.s1 proof defpred P[Nat] means r in F.$1; r in F.s by A10,A11; then A12:ex k being Nat st P[k]; ex k being Nat st P[k] & for n being Nat st P[n] holds k <= n from Min(A12); then consider k being Nat such that A13:r in F.k & for n being Nat st r in F.n holds k <= n; A14:k=0 implies ex s1 being Element of NAT st r in H.s1 by A2,A13; (ex l being Nat st k = l + 1) implies (ex s1 being Element of NAT st r in H.s1) proof given l being Nat such that A15:k = l + 1; A16:not r in G.l proof assume r in G.l; then consider i being Nat such that A17:i <= l & r in F.i by A1,MEASURE2:6; k <= i by A13,A17; hence thesis by A15,A17,NAT_1:38; end; A18: H.(l+1) = F.(l+1) \ G.l by A2; take k; thus thesis by A13,A15,A16,A18,XBOOLE_0:def 4; end; hence thesis by A14,NAT_1:22; end; then consider s1 being Element of NAT such that A19:r in H.s1; H.s1 in rng H by FUNCT_2:6; hence thesis by A19,TARSKI:def 4; end; thus thesis by A6; end; theorem Th6: bool X is sigma_Field_Subset of X proof A1:for A being set holds A in bool X implies X\A in bool X proof let A be set; assume A in bool X; X \ A c= X by XBOOLE_1:36; hence thesis; end; for M being N_Sub_set_fam of X st M c= bool X holds union M in bool X; hence thesis by A1,MEASURE1:def 3,def 9,SETFAM_1:def 7; end; definition let X be set; let F be Function of NAT,bool X; redefine func rng F -> Subset-Family of X; coherence proof rng F c= bool X by RELSET_1:12; hence thesis by SETFAM_1:def 7; end; end; definition let X be set; let A be Subset-Family of X; redefine func union A -> Element of bool X; coherence proof union A c= union bool X by ZFMISC_1:95; hence thesis; end; end; definition let Y be set; let X,Z be non empty set; let F be Function of Y,X; let M be Function of X,Z; redefine func M*F -> Function of Y,Z; coherence proof thus M*F is Function of Y,Z; end; end; theorem Th7: for a,b being R_eal holds ex M being Function of bool X,ExtREAL st for A being Element of bool X holds ((A = {} implies M.A = a) & (A <> {} implies M.A = b)) proof let a,b be R_eal; defpred P[set,set] means (($1 = {} implies $2 = a) & ($1 <> {} implies $2 = b)); A1:for x being set st x in bool X ex y being set st y in ExtREAL & P[x,y] proof let x be set; x <> {} implies ex y being set st y in ExtREAL & P[x,y]; hence thesis; end; ex F being Function of bool X,ExtREAL st for x being set st x in bool X holds P[x,F.x] from FuncEx1(A1); then consider M being Function of bool X,ExtREAL such that A2:for x being set st x in bool X holds P[x,M.x]; take M; thus thesis by A2; end; theorem Th8: ex M being Function of bool X,ExtREAL st for A being Element of bool X holds M.A = 0. proof consider M being Function of bool X,ExtREAL such that A1:for A being Element of bool X holds (A = {} implies M.A = 0.) & (A <> {} implies M.A = 0.) by Th7; A2:for A being Element of bool X holds M.A = 0. proof let A be Element of bool X; A = {} implies M.A = 0.by A1; hence thesis by A1; end; take M; thus thesis by A2; end; canceled 2; theorem Th11: ex M being Function of bool X,ExtREAL st M is nonnegative & M.{} = 0. & for A,B being Element of bool X st A c= B holds M.A <=' M.B & for F being Function of NAT,bool X holds M.(union rng F) <=' SUM(M*F) proof consider M being Function of bool X,ExtREAL such that A1:for A being Element of bool X holds M.A = 0.by Th8; A2:for A being Element of bool X holds 0.<=' M.A by A1; then A3:M is nonnegative by MEASURE1:def 4; A4:{} c= X by XBOOLE_1:2; A5:for F being Function of NAT,bool X holds M.(union rng F) <=' SUM(M*F) proof let F be Function of NAT,bool X; A6:M.(union rng F) = 0.by A1; A7:M*F is nonnegative by A3,MEASURE1:54; A8:for r being Nat st 0 <= r holds (M*F).r = 0. proof let r be Nat; dom (M*F) = NAT by FUNCT_2:def 1; then (M*F).r = M.(F.r) by FUNCT_1:22; hence thesis by A1; end; then A9:SUM(M*F) = Ser(M*F).0 by A7,SUPINF_2:67; Ser(M*F).0 = (M*F).0 by SUPINF_2:63; hence thesis by A6,A8,A9; end; take M; for A,B being Element of bool X holds M.A <=' M.B proof let A,B be Element of bool X; M.A = 0. & M.B = 0. by A1; hence thesis; end; hence thesis by A1,A2,A4,A5,MEASURE1:def 4; end; definition let X be set; canceled; mode C_Measure of X -> Function of bool X,ExtREAL means :Def2:it is nonnegative & it.{} = 0. & for A,B being Element of bool X st A c= B holds it.A <=' it.B & for F being Function of NAT,bool X holds it.(union rng F) <=' SUM(it*F); existence by Th11; end; reserve C for C_Measure of X; definition let X be set; let C be C_Measure of X; func sigma_Field(C) -> non empty Subset-Family of X means :Def3:for A being Element of bool X holds (A in it iff for W,Z being Element of bool X holds (W c= A & Z c= X \ A implies C.W + C.Z <=' C.(W \/ Z))); existence proof defpred P[set] means for A being set holds (A = $1 implies for W,Z being Element of bool X holds (W c= A & Z c= X \ A implies C.W + C.Z <=' C.(W \/ Z))); consider D being set such that A1:for y being set holds (y in D iff (y in bool X & P[y])) from Separation; A2:for A being set holds (A in D iff A in bool X & for W,Z being Element of bool X holds (W c= A & Z c= X \ A implies C.W + C.Z <=' C.(W \/ Z))) proof let A be set; P[A] iff for W,Z being Element of bool X holds (W c= A & Z c= X \ A implies C.W + C.Z <=' C.(W \/ Z)); hence thesis by A1; end; reconsider A = {} as Element of bool X by XBOOLE_1:2; A3: now let W,Z be Element of bool X; assume W c= A & Z c= X \ A; then A4:W = {} by XBOOLE_1:3; then C.W = 0. by Def2; hence C.W + C.Z <=' C.(W \/ Z) by A4,SUPINF_2:18; end; then D <> {} & for A being set holds A in D implies A in bool X by A2; then D c= bool X by TARSKI:def 3; then reconsider D as non empty Subset-Family of X by A2,A3,SETFAM_1:def 7; take D; thus thesis by A2; end; uniqueness proof let D1,D2 be non empty Subset-Family of X such that A5:for A being Element of bool X holds (A in D1 iff for W,Z being Element of bool X holds (W c= A & Z c= X \ A implies C.W + C.Z <=' C.(W \/ Z))) and A6:for A being Element of bool X holds (A in D2 iff for W,Z being Element of bool X holds (W c= A & Z c= X \ A implies C.W + C.Z <=' C.(W \/ Z))); for A being set holds A in D1 iff A in D2 proof let A be set; hereby assume A7:A in D1; then reconsider A' = A as Element of bool X; for W,Z being Element of bool X holds (W c= A & Z c= X \ A' implies C.W + C.Z <=' C.(W \/ Z)) by A5,A7; hence A in D2 by A6; end; assume A8:A in D2; then reconsider A as Element of bool X; for W,Z being Element of bool X holds (W c= A & Z c= X \ A implies C.W + C.Z <=' C.(W \/ Z)) by A6,A8; hence thesis by A5; end; hence thesis by TARSKI:2; end; end; theorem Th12: for W,Z being Element of bool X holds C.(W \/ Z) <=' C.W + C.Z proof let W,Z be Element of bool X; reconsider P = {} as Subset of X by XBOOLE_1:2; consider F being Function of NAT,bool X such that A1:rng F = {W,Z,P} & F.0 = W & F.1 = Z & for n being Nat st 1 < n holds F.n = P by MEASURE1:38; A2: union {W,Z,P} = W \/ Z proof thus union {W,Z,P} c= W \/ Z proof let x be set; assume x in union {W,Z,P}; then consider Y being set such that A3:x in Y & Y in {W,Z,P} by TARSKI:def 4; x in W or x in Z or x in P by A3,ENUMSET1:13; hence thesis by XBOOLE_0:def 2; end; let x be set; assume A4:x in W \/ Z; now per cases by A4,XBOOLE_0:def 2; suppose A5:x in W; take Y = W; thus x in Y & Y in {W,Z,P} by A5,ENUMSET1:14; suppose A6:x in Z; take Y = Z; thus x in Y & Y in {W,Z,P} by A6,ENUMSET1:14; end; hence thesis by TARSKI:def 4; end; C is nonnegative by Def2; then A7:C*F is nonnegative by MEASURE1:54; A8:for r being Nat st 2 <= r holds (C*F).r = 0. proof let r be Nat; assume 2 <= r; then 1 + 1 <= r; then 1 < r by NAT_1:38; then F.r = {} by A1; then C.(F.r) = 0. by Def2; hence thesis by FUNCT_2:21; end; F.0 = W & F.1 = Z & F.2 = P by A1; then A9:(C*F).2 = C.P & (C*F).1 = C.Z & (C*F).0 = C.W by FUNCT_2:21; set G = C*F; consider y1,y2 being R_eal such that A10:y1 = Ser(G).1 & y2 = Ser(G).0; Ser(G).2 = y1 + G.(1 + 1) by A10,SUPINF_2:63 .= Ser(G).1 + 0. by A9,A10,Def2 .= Ser(G).1 by SUPINF_2:18 .= y2 + G.(0 + 1) by A10,SUPINF_2:63 .= C.W + C.Z by A9,A10,SUPINF_2:63; then SUM(C*F) = C.W + C.Z by A7,A8,SUPINF_2:67; hence thesis by A1,A2,Def2; end; theorem for W,Z being Element of bool X holds C.Z + C.W = C.W + C.Z; theorem Th14: for A being Element of bool X holds (A in sigma_Field(C) iff for W,Z being Element of bool X holds (W c= A & Z c= X \ A implies C.W + C.Z = C.(W \/ Z))) proof let A be Element of bool X; hereby assume A1: A in sigma_Field(C); let W,Z be Element of bool X; assume W c= A & Z c= X \ A; then C.(W \/ Z) <=' C.W + C.Z & C.W + C.Z <=' C.(W \/ Z) by A1,Def3,Th12 ; hence C.W + C.Z = C.(W \/ Z) by SUPINF_1:22; end; assume for W,Z being Element of bool X holds (W c= A & Z c= X \ A implies C.W + C.Z = C.(W \/ Z)); then for W,Z being Element of bool X holds (W c= A & Z c= X \ A implies C.W + C.Z <=' C.(W \/ Z)); hence thesis by Def3; end; theorem Th15: for W,Z being Element of bool X holds (W in sigma_Field(C) & Z in sigma_Field(C) & Z misses W) implies C.(W \/ Z) = C.W + C.Z proof let W,Z be Element of bool X; assume A1:W in sigma_Field(C) & Z in sigma_Field(C) & Z misses W; Z \ W c= X \ W by XBOOLE_1:33; then Z c= X \ W by A1,XBOOLE_1:83; hence thesis by A1,Th14; end; theorem Th16: A in sigma_Field(C) implies X \ A in sigma_Field(C) proof assume A1:A in sigma_Field(C); A2:X \ A c= X by XBOOLE_1:109; for W,Z being Element of bool X holds W c= X \ A & Z c= X \ (X \ A) implies C.W + C.Z <=' C.(W \/ Z) proof let W,Z be Element of bool X; assume A3:W c= X \ A & Z c= X \ (X \ A); X \ (X \ A) = X /\ A by XBOOLE_1:48; then Z c= A & W c= X \ A by A1,A3,XBOOLE_1:28; hence thesis by A1,Def3; end; hence thesis by A2,Def3; end; theorem Th17: A in sigma_Field(C) & B in sigma_Field(C) implies A \/ B in sigma_Field(C) proof assume A1:A in sigma_Field(C) & B in sigma_Field(C); then reconsider A,B as Element of bool X; set D = A \/ B; for W,Z being Element of bool X holds W c= D & Z c= X \ D implies C.W + C.Z = C.(W \/ Z) proof let W,Z be Element of bool X; assume A2: W c= D & Z c= X \ D; set W1 = W /\ A; A3:W \ A c= X \ A by XBOOLE_1:33; set W2 = W \ A; (X \ A) /\ (X \ B) c= X \ A by XBOOLE_1:17; then X \ (A \/ B) c= X \ A by XBOOLE_1:53; then A4:W1 c= A & Z c= X \ A by A2,XBOOLE_1:1,17; A5: W2 \/ Z c= X \ A \/ Z by A3,XBOOLE_1:9; set Z1 = W2 \/ Z; A6:Z1 c= X \ A by A4,A5,XBOOLE_1:12; A7:W = W1 \/ W2 by XBOOLE_1:51; A8:C.(W \/ Z) = C.((W1 \/ W2) \/ Z) by XBOOLE_1:51 .= C.(W1 \/ Z1) by XBOOLE_1:4 .= C.W1 + C.Z1 by A1,A4,A6,Th14; W \ A c= B \/ A \ A by A2,XBOOLE_1:33; then W \ A c= B \ A & B \ A c= B by XBOOLE_1:36,40; then A9:W2 c= B by XBOOLE_1:1; Z c= (X \ A) /\ (X \ B) & (X \ A) /\ (X \ B) c= X \ B by A2,XBOOLE_1:17,53; then Z c= X \ B by XBOOLE_1:1; then A10:C.W2 + C.Z <=' C.Z1 by A1,A9,Def3; C is nonnegative by Def2; then A11:0.<=' C.W2 & 0.<=' C.Z by MEASURE1:def 4; then 0. + 0. <=' C.W2 + C.Z by MEASURE1:4; then A12:0. <=' C.W2 + C.Z by SUPINF_2:18; C is nonnegative by Def2; then A13:0. <=' C.W1 by MEASURE1:def 4; then C.W1 + (C.W2 + C.Z) <=' C.(W \/ Z) by A8,A10,A12,MEASURE1:4; then A14:(C.W1 + C.W2) + C.Z <=' C.(W \/ Z) by A11,A13,Th1; C is nonnegative by Def2; then A15:0.<=' C.W & 0.<=' C.Z & C.Z <=' C.Z by MEASURE1:def 4; C.W <=' C.W1 + C.W2 by A7,Th12; then C.W + C.Z <=' (C.W1 + C.W2) + C.Z by A15,MEASURE1:4; then A16:C.W + C.Z <=' C.(W \/ Z) by A14,SUPINF_1:29; C.(W \/ Z) <=' C.W + C.Z by Th12; hence thesis by A16,SUPINF_1:22; end; hence thesis by Th14; end; theorem Th18: A in sigma_Field(C) & B in sigma_Field(C) implies A /\ B in sigma_Field(C) proof assume A1:A in sigma_Field(C) & B in sigma_Field(C); then A /\ B c= X /\ X by XBOOLE_1:27; then A2:A /\ B = X /\ (A /\ B) by XBOOLE_1:28 .= X \ (X \(A /\ B)) by XBOOLE_1:48 .= X \ ((X \ A) \/ (X \ B)) by XBOOLE_1:54; X \ A in sigma_Field(C) & X \ B in sigma_Field(C) by A1,Th16; then (X \ A) \/ (X \ B) in sigma_Field(C) by Th17; hence thesis by A2,Th16; end; theorem Th19: A in sigma_Field(C) & B in sigma_Field(C) implies A \ B in sigma_Field(C) proof assume A1:A in sigma_Field(C) & B in sigma_Field(C); for x being set holds x in A \ B iff x in A /\ (X \ B) proof let x be set; hereby assume x in A \ B; then x in A & not x in B by XBOOLE_0:def 4; then x in A & x in (X \ B) by A1,XBOOLE_0:def 4; hence x in A /\ (X \ B) by XBOOLE_0:def 3; end; assume x in A /\ (X \ B); then x in A & x in (X \ B) by XBOOLE_0:def 3; then x in A & (x in X & not x in B) by XBOOLE_0:def 4; hence thesis by XBOOLE_0:def 4; end; then A2:A \ B = A /\ (X \ B) by TARSKI:2; X \ B in sigma_Field(C) by A1,Th16; hence thesis by A1,A2,Th18; end; theorem Th20: for N being Function of NAT,S holds for A being Element of S holds ex F being Function of NAT,S st for n being Element of NAT holds F.n = A /\ N.n proof let N be Function of NAT,S; let A be Element of S; defpred P[set,set] means ($1 in NAT implies $2 = A /\ N.$1); A1:for x being set st x in NAT ex y being set st y in S & P[x,y] proof let x be set; assume x in NAT; then reconsider x as Element of NAT; consider y being set such that A2:y = A /\ N.x; take y; thus thesis by A2; end; ex F being Function of NAT,S st for x being set st x in NAT holds P[x,F.x] from FuncEx1(A1); then consider F being Function of NAT,S such that A3:for x being set st x in NAT holds P[x,F.x]; take F; thus thesis by A3; end; theorem Th21: sigma_Field(C) is sigma_Field_Subset of X proof A1:for A being set holds A in sigma_Field(C) implies X\A in sigma_Field(C) by Th16; A2:C is nonnegative by Def2; for M being N_Sub_set_fam of X holds M c= sigma_Field(C) implies union M in sigma_Field(C) proof let M be N_Sub_set_fam of X; assume A3:M c= sigma_Field(C); for W,Z being Element of bool X holds (W c= union M & Z c= X \ union M implies C.W + C.Z <=' C.(W \/ Z)) proof let W,Z be Element of bool X; assume A4:W c= union M & Z c= X \ union M; reconsider S = bool X as sigma_Field_Subset of X by Th6; consider F being Function of NAT,bool X such that A5:rng F = M by SUPINF_2:def 14; A6:for n being Element of NAT holds F.n in sigma_Field(C) proof let n be Element of NAT; F.n in M by A5,FUNCT_2:6; hence thesis by A3; end; consider G being Function of NAT,S such that A7:G.0 = F.0 & for n being Element of NAT holds G.(n+1) = F.(n+1) \/ G.n by MEASURE2:5; defpred P[Nat] means G.$1 in sigma_Field(C); A8: P[0] by A6,A7; A9: for k being Nat st P[k] holds P[k+1] proof let k be Nat; assume A10:G.k in sigma_Field(C); A11:G.(k+1) = F.(k+1) \/ G.k by A7; F.(k+1) in sigma_Field(C) by A6; hence thesis by A10,A11,Th17; end; A12: for n being Element of NAT holds P[n] from Ind(A8,A9); consider B being Function of NAT,S such that A13:B.0 = F.0 & for n being Element of NAT holds B.(n+1) = F.(n+1) \ G.n by MEASURE2:9; A14:union rng F = union rng B by A7,A13,Th5; consider Q being Function of NAT,S such that A15:for n being Element of NAT holds Q.n = W /\ B.n by Th20; A16:union rng Q = W /\ union rng B by A15,Th4; consider QQ being Function of NAT,S such that A17:QQ.0 = Q.0 & for n being Element of NAT holds QQ.(n+1) = Q.(n+1) \/ QQ.n by MEASURE2:5; reconsider Q,QQ,F,G as Function of NAT,bool X; A18: C*Q is nonnegative by A2,MEASURE1:54; A19:QQ.0 = W /\ F.0 by A13,A15,A17; A20:F.0 in sigma_Field(C) by A6; A21:QQ.0 c= F.0 by A19,XBOOLE_1:17; defpred P[Nat] means C.(Z \/ QQ.$1) = C.Z + Ser(C*Q).$1; F.0 in rng F by FUNCT_2:6; then F.0 c= union rng F by ZFMISC_1:92; then X \ union rng F c= X \ F.0 by XBOOLE_1:34; then A22: Z c= X \ F.0 by A4,A5,XBOOLE_1:1; Ser(C*Q).0 = (C*Q).0 by SUPINF_2:63 .= C.(QQ.0) by A17,FUNCT_2:21; then A23: P[0] by A20,A21,A22,Th14; A24: for k being Nat st P[k] holds P[k+1] proof let k be Nat; assume A25:C.(Z \/ QQ.k) = C.Z + Ser(C*Q).k; A26: QQ.(k+1) = QQ.k \/ Q.(k+1) by A17; Q.(k+1) = W /\ B.(k+1) by A15 .= W /\ (F.(k+1) \ G.k) by A13; then A27:Q.(k+1) c= F.(k+1) \ G.k by XBOOLE_1:17; F.(k+1) in sigma_Field(C) & G.k in sigma_Field(C) by A6,A12; then A28:F.(k+1) \ G.k in sigma_Field(C) by Th19; F.(k+1) in rng F by FUNCT_2:6; then A29:F.(k+1) c= union rng F by ZFMISC_1:92; F.(k+1) \ G.k c= F.(k+1) by XBOOLE_1:36; then F.(k+1) \ G.k c= union rng F by A29,XBOOLE_1:1; then X \ union rng F c= X \ (F.(k+1) \ G.k) by XBOOLE_1:34; then A30:Z c= X \ (F.(k+1) \ G.k) by A4,A5,XBOOLE_1:1; defpred P[Nat] means QQ.$1 c= G.$1; QQ.0 = W /\ F.0 by A13,A15,A17; then A31: P[0] by A7,XBOOLE_1:17; for n being Nat holds QQ.n misses (F.(n+1) \ G.n) proof let n be Nat; A32: for n being Nat st P[n] holds P[n+1] proof let n be Nat; assume A33:QQ.n c= G.n; A34:QQ.(n+1) = Q.(n+1) \/ QQ.n by A17 .= (W /\ B.(n+1)) \/ QQ.n by A15 .= (W /\ (F.(n+1) \ G.n)) \/ QQ.n by A13; F.(n+1) \ G.n c= F.(n+1) by XBOOLE_1:36; then A35:W /\ (F.(n+1) \ G.n) c= W /\ F.(n+1) by XBOOLE_1:26; W /\ F.(n+1) c= F.(n+1) by XBOOLE_1:17; then W /\ (F.(n+1) \ G.n) c= F.(n+1) by A35,XBOOLE_1:1; then QQ.(n+1) c= F.(n+1) \/ G.n by A33,A34,XBOOLE_1:13; hence QQ.(n+1) c= G.(n+1) by A7; end; for n being Nat holds P[n] from Ind(A31,A32); then A36:QQ.n c= G.n; G.n misses (F.(n+1) \ G.n) by XBOOLE_1:79; hence thesis by A36,XBOOLE_1:63; end; then QQ.k misses (F.(k+1) \ G.k); then A37:QQ.k /\ (F.(k+1) \ G.k) = {} by XBOOLE_0:def 7; QQ.k c= X \ (F.(k+1) \ G.k) proof let z be set; assume A38:z in QQ.k; then not z in F.(k+1) \ G.k by A37,XBOOLE_0:def 3; hence thesis by A38,XBOOLE_0:def 4; end; then A39: Z \/ QQ.k c= X \ (F.(k+1) \ G.k) by A30,XBOOLE_1:8; A40:0. <=' C.Z & 0. <=' Ser(C*Q).k & 0. <=' (C*Q).(k+1) by A2,A18,SUPINF_2:58,59; C.(Q.(k+1)) + C.(Z \/ QQ.k) = C.((Z \/ QQ.k) \/ Q.(k+1)) by A27, A28,A39,Th14 .= C.(Z \/ QQ.(k+1)) by A26,XBOOLE_1:4; then C.(Z \/ QQ.(k+1)) = (C.Z + Ser(C*Q).k) + (C*Q).(k+1) by A25, FUNCT_2:21 .= C.Z + (Ser(C*Q).k + (C*Q).(k+1)) by A40,Th1 .= C.Z + Ser(C*Q).(k+1) by SUPINF_2:63; hence thesis; end; A41:for n being Element of NAT holds P[n] from Ind(A23,A24); defpred Q[Nat] means QQ.$1 c= W; QQ.0 = W /\ B.0 by A15,A17; then A42: Q[0] by XBOOLE_1:17; A43: for n being Nat st Q[n] holds Q[n+1] proof let n be Nat; assume A44:QQ.n c= W; A45:QQ.(n+1) = Q.(n+1) \/ QQ.n by A17 .= (W /\ B.(n+1)) \/ QQ.n by A15; W /\ B.(n+1) c= W by XBOOLE_1:17; then QQ.(n+1) c= W \/ W by A44,A45,XBOOLE_1:13; hence thesis; end; A46: for n being Nat holds Q[n] from Ind(A42,A43); A47:union rng Q = W by A4,A5,A14,A16,XBOOLE_1:28; A48:for n being Element of NAT holds Ser(C*Q).n + C.Z <=' C.(Z \/ W) proof let n be Element of NAT; A49:Ser(C*Q).n + C.Z = C.(Z \/ QQ.n) by A41; QQ.n c= W by A46; then Z \/ QQ.n c= Z \/ W by XBOOLE_1:9; hence thesis by A49,Def2; end; A50:C.Z = +infty implies C.W + C.Z <=' C.(W \/ Z) proof assume A51:C.Z = +infty; 0. <=' C.W by A2,MEASURE1:def 4; then A52:C.W + C.Z = +infty by A51,SUPINF_2:19,def 2; Z c= W \/ Z by XBOOLE_1:7; hence thesis by A51,A52,Def2; end; (not C.Z = -infty & not C.Z = +infty) implies C.W + C.Z <=' C.(W \/ Z ) proof assume A53:not C.Z = -infty & not C.Z = +infty; A54:for n being Element of NAT holds Ser(C*Q).n <=' C.(Z \/ W) - C.Z proof let n be Element of NAT; Ser(C*Q).n + C.Z <=' C.(Z \/ W) by A48; hence thesis by A53,Th2; end; defpred P[set,set] means ($1 = 0 implies $2 = C.(Z \/ W) - C.Z ) & ($1 <> 0 implies $2 = 0.); A55:for x being set st x in NAT ex y being set st y in ExtREAL & P[x,y] proof let x be set; assume x in NAT; then reconsider x as Element of NAT; x <> 0 implies ex y being set st y in ExtREAL & P[x,y]; hence thesis; end; ex R being Function of NAT,ExtREAL st for x being set st x in NAT holds P[x,R.x] from FuncEx1(A55); then consider R being Function of NAT,ExtREAL such that A56:for x being set st x in NAT holds P[x,R.x]; for n being Nat holds 0. <=' R.n proof let n be Nat; per cases; suppose A57:n = 0; Z c= Z \/ W by XBOOLE_1:7; then C.Z <=' C.(Z \/ W) by Def2; then A58:C.Z - C.Z <=' C.(Z \/ W) - C.Z by A53,SUPINF_2:15; C.Z in REAL or C.Z in {-infty,+infty} by SUPINF_1:def 6,XBOOLE_0:def 2; then consider y being Real such that A59:y = C.Z by A53,TARSKI:def 2; A60:C.Z - C.Z = y - y by A59,SUPINF_2:5; y - y = 0 by XCMPLX_1:14; hence thesis by A56,A57,A58,A60,SUPINF_2:def 1; suppose n <> 0; hence thesis by A56; end; then A61:R is nonnegative by SUPINF_2:58; A62: for n being Nat holds Ser(C*Q).n <=' Ser(R).n proof defpred P[Nat] means Ser(R).$1 = C.(Z \/ W) - C.Z; let n be Nat; Ser(R).0 = R.0 by SUPINF_2:63; then A63: P[0] by A56; A64: for k being Nat st P[k] holds P[k+1] proof let k be Nat; assume A65:Ser(R).k = C.(Z \/ W) - C.Z; set y = Ser(R).k; thus Ser(R).(k+1) = y + R.(k+1) by SUPINF_2:63 .= y + 0. by A56 .= C.(Z \/ W) - C.Z by A65,SUPINF_2:18; end; for k being Nat holds P[k] from Ind(A63,A64); then Ser(R).n = C.(Z \/ W) - C.Z; hence thesis by A54; end; now let k be Nat; assume 1 <= k; then k <> 0; hence R.k = 0. by A56; end; then A66:SUM(R) = Ser(R).1 by A61,SUPINF_2:67; set y = Ser(R).0; y = R.0 by SUPINF_2:63; then A67:y = C.(Z \/ W) - C.Z by A56; Ser(R).1 = y + R.(0+1) by SUPINF_2:63 .= y + 0. by A56 .= C.(Z \/ W) - C.Z by A67,SUPINF_2:18; then A68:SUM(C*Q) <=' C.(Z \/ W) - C.Z by A62,A66,MEASURE3:3; C.W <=' SUM(C*Q) by A47,Def2; then C.W <=' C.(Z \/ W) - C.Z by A68,SUPINF_1:29; hence thesis by A53,Th2; end; hence thesis by A2,A50,MEASURE1:def 4,SUPINF_2:19; end; hence thesis by Def3; end; hence thesis by A1,MEASURE1:def 3,def 9; end; definition let X be set; let C be C_Measure of X; cluster sigma_Field(C) -> sigma_Field_Subset-like compl-closed non empty; coherence by Th21; end; definition let X be set; let S be sigma_Field_Subset of X; let A be N_Sub_fam of S; redefine func union A -> Element of S; coherence proof A c= S by MEASURE2:def 1; hence thesis by MEASURE1:def 9; end; end; definition let X be set; let C be C_Measure of X; func sigma_Meas(C) -> Function of sigma_Field(C),ExtREAL means :Def4:for A being Element of bool X st A in sigma_Field(C) holds it.A = C.A; existence proof ex D being Function of sigma_Field(C),ExtREAL st (for A being Element of bool X holds A in sigma_Field(C) implies D.A = C.A) proof deffunc F(set) = C.$1; A1:for S being set st S in sigma_Field(C) holds (F(S) in ExtREAL) by FUNCT_2:7; consider D being Function of sigma_Field(C),ExtREAL such that A2:for S being set st S in sigma_Field(C) holds D.S = F(S) from Lambda1(A1); take D; thus thesis by A2; end; then consider D being Function of sigma_Field(C),ExtREAL such that A3:for A being Element of bool X st A in sigma_Field(C) holds D.A = C.A; take D; thus thesis by A3; end; uniqueness proof let D1,D2 be Function of sigma_Field(C),ExtREAL such that A4:for A being Element of bool X holds A in sigma_Field(C) implies D1.A = C.A and A5:for A being Element of bool X holds A in sigma_Field(C) implies D2.A = C.A; A6:(sigma_Field(C) = dom D1) & (sigma_Field(C) = dom D2) by FUNCT_2:def 1; for A being set st A in sigma_Field(C) holds D1.A = D2.A proof let A be set; assume A7:A in sigma_Field(C); then reconsider A as Element of bool X; D1.A = C.A by A4,A7 .= D2.A by A5,A7; hence thesis; end; hence thesis by A6,FUNCT_1:9; end; end; theorem Th22: sigma_Meas(C) is Measure of sigma_Field(C) proof A1:now let A be Element of sigma_Field(C); reconsider A' = A as Element of bool X; A2: (sigma_Meas(C)).A' = C.A' by Def4; C is nonnegative by Def2; hence 0.<=' (sigma_Meas(C)).A by A2,MEASURE1:def 4; end; {} in sigma_Field(C) by MEASURE1:45; then (sigma_Meas(C)).{} = C.{} by Def4; then A3:sigma_Meas(C) is nonnegative & (sigma_Meas(C)).{} = 0. by A1,Def2,MEASURE1:def 4; now let A,B be Element of sigma_Field(C); assume A4: A misses B; reconsider A' = A,B' = B as Element of bool X; A5:(sigma_Meas(C)).A' = C.A' & (sigma_Meas(C)).B' = C.B' by Def4; C.(A' \/ B') = C.A' + C.B' by A4,Th15; hence (sigma_Meas(C)).(A \/ B) = (sigma_Meas(C)).A + (sigma_Meas(C)).B by A5,Def4; end; hence thesis by A3,MEASURE1:def 5; end; definition let X be set; let C be C_Measure of X; let A be Element of sigma_Field(C); redefine func C.A -> R_eal; coherence by FUNCT_2:7; end; theorem Th23: sigma_Meas(C) is sigma_Measure of sigma_Field(C) proof reconsider M = sigma_Meas(C) as Measure of sigma_Field(C) by Th22; for F being Sep_Sequence of sigma_Field(C) holds M.(union rng F) <=' SUM(M*F) proof let F be Sep_Sequence of sigma_Field(C); consider A being Element of bool X such that A1:A = union rng F; M.A in ExtREAL by A1,FUNCT_2:7; then consider a,b being R_eal such that A2:a = M.A & b = C.A; reconsider F' = F as Function of NAT,bool X by FUNCT_2:9; A3: C*F' is Function of NAT,ExtREAL; for k being set st k in NAT holds (M*F).k = (C*F).k proof let k be set; assume A4:k in NAT; then A5:(M*F).k = M.(F.k) by FUNCT_2:21; A6:F.k in sigma_Field(C) by A4,FUNCT_2:7; reconsider F as Function of NAT,bool X by FUNCT_2:9; (C*F).k = C.(F.k) by A4,FUNCT_2:21; hence thesis by A5,A6,Def4; end; then A7: M*F = C*F by A3,FUNCT_2:18; reconsider F as Function of NAT,bool X by FUNCT_2:9; b <=' SUM(C*F) by A1,A2,Def2; hence thesis by A1,A2,A7,Def4; end; hence thesis by MEASURE3:17; end; definition let X be set; let C be C_Measure of X; redefine func sigma_Meas(C) -> sigma_Measure of sigma_Field(C); coherence by Th23; end; theorem Th24: for A being Element of bool X holds C.A = 0. implies A in sigma_Field(C) proof let A be Element of bool X; assume A1:C.A = 0.; now let W,Z be Element of bool X; assume A2:W c= A & Z c= X \ A; C is nonnegative by Def2; then 0.<=' C.W & C.W <=' 0. by A1,A2,Def2,MEASURE1:def 4; then A3:C.W = 0. by SUPINF_1:22; Z c= W \/ Z by XBOOLE_1:7; then C.Z <=' C.(W \/ Z) by Def2; hence C.W + C.Z <=' C.(W \/ Z) by A3,SUPINF_2:18; end; hence thesis by Def3; end; theorem sigma_Meas(C) is_complete sigma_Field(C) proof let A be Subset of X; let B; assume that A1:B in sigma_Field(C) and A2:A c= B & (sigma_Meas(C)).B = 0.; reconsider B as Element of bool X by A1; A3:C.B = 0. by A1,A2,Def4; C is nonnegative by Def2; then 0.<=' C.A & C.A <=' 0. by A2,A3,Def2,MEASURE1:def 4; then C.A = 0. by SUPINF_1:22; hence thesis by Th24; end;