### The Mizar article:

### The $\sigma$-additive Measure Theory

**by****Jozef Bialas**

- Received October 15, 1990
Copyright (c) 1990 Association of Mizar Users

- MML identifier: MEASURE1
- [ MML identifier index ]

environ vocabulary TARSKI, BOOLE, SUPINF_1, RLVECT_1, ARYTM_3, ARYTM_1, SETFAM_1, PROB_1, SUBSET_1, FINSUB_1, FUNCT_1, SUPINF_2, RELAT_1, FUNCOP_1, CARD_4, PROB_2, MEASURE1; notation TARSKI, XBOOLE_0, ENUMSET1, ZFMISC_1, SUBSET_1, NUMBERS, XREAL_0, REAL_1, RELAT_1, FINSUB_1, FUNCT_1, FUNCT_2, NAT_1, SETFAM_1, FUNCOP_1, PROB_1, PROB_2, SUPINF_1, SUPINF_2; constructors ENUMSET1, NAT_1, REAL_1, SUPINF_2, FUNCOP_1, FINSUB_1, PROB_2, MEMBERED, XBOOLE_0; clusters SUBSET_1, SUPINF_1, RELSET_1, ARYTM_3, PROB_1, MEMBERED, ZFMISC_1, XBOOLE_0, NUMBERS, ORDINAL2; requirements NUMERALS, REAL, SUBSET, BOOLE, ARITHM; definitions SUPINF_2, TARSKI, FINSUB_1, PROB_2, PROB_1, XBOOLE_0; theorems TARSKI, FUNCT_1, FUNCT_2, ENUMSET1, ZFMISC_1, NAT_1, SETFAM_1, SUPINF_1, SUPINF_2, CQC_THE1, RELAT_1, RELSET_1, FUNCOP_1, FINSUB_1, PROB_2, PROB_1, SUBSET_1, XBOOLE_0, XBOOLE_1, XCMPLX_0, XCMPLX_1; schemes FUNCT_2, XBOOLE_0; begin :: Some useful theorems about sets and R_eal numbers reserve X for set; theorem Th1: for X,Y being set holds union {X,Y,{}} = union {X,Y} proof let X,Y be set; thus union {X,Y,{}} = union ({X,Y} \/ {{}}) by ENUMSET1:43 .= union {X,Y} \/ union {{}} by ZFMISC_1:96 .= union {X,Y} \/ {} by ZFMISC_1:31 .= union {X,Y}; end; canceled 2; theorem Th4: for x,y,s,t being R_eal holds 0.<=' x & 0.<=' s & x <=' y & s <=' t implies x + s <=' y + t proof let x,y,s,t be R_eal; assume A1:0. <=' x & 0.<=' s & x <=' y & s <=' t; not ((x = +infty & s = -infty) or (x = -infty & s = +infty)) & not ((y = +infty & t = -infty) or (y = -infty & t = +infty)) proof 0.<=' y & 0.<=' t by A1,SUPINF_1:29; hence thesis by A1,SUPINF_1:17,SUPINF_2:def 1; end; hence thesis by A1,SUPINF_2:14; end; theorem Th5: for x,y,z being R_eal holds 0.<=' y & 0.<=' z & x = y + z & y <' +infty implies z = x - y proof let x,y,z be R_eal; assume A1:0. <=' y & 0.<=' z & x = y + z & y <' +infty; then 0. + 0. <=' y + z by Th4; then A2:0. <=' x by A1,SUPINF_2:18; (y + z) - y = z proof A3:not x = -infty & not y = -infty & not z = -infty by A1,A2,SUPINF_1:17,SUPINF_2:def 1; x in REAL \/ {-infty,+infty} & y in REAL \/ {-infty,+infty} by SUPINF_1:def 5; then A4: (x in REAL or x in {-infty,+infty}) & (y in REAL or y in {-infty,+infty}) by XBOOLE_0:def 2; A5: (x in REAL & y in REAL) implies (y + z) - y = z proof assume x in REAL & y in REAL; then reconsider a = y, b = z as Real by A1,A3,SUPINF_2:12; y + z = a + b by SUPINF_2:1; then (y + z) - y = (a + b) - a by SUPINF_2:5 .= (b - a) + a by XCMPLX_1:29 .= (b + (-a)) + a by XCMPLX_0:def 8 .= b + ((-a) + a) by XCMPLX_1:1 .= b + 0 by XCMPLX_0:def 6 .= z; hence thesis; end; (x = +infty & y in REAL) implies (y + z) - y = z proof assume A6:x = +infty & y in REAL; then (y + z) - y = +infty by A1,SUPINF_2:6 .= z by A1,A6,SUPINF_2:8; hence thesis; end; hence thesis by A1,A3,A4,A5,TARSKI:def 2; end; hence thesis by A1; end; theorem for A being Subset of X holds {A} is Subset-Family of X proof let A be Subset of X; {A} c= bool X by ZFMISC_1:37; hence thesis by SETFAM_1:def 7; end; theorem Th7: for A,B being Subset of X holds {A,B} is Subset-Family of X proof let A,B be Subset of X; set C = {A,B}; C c= bool X proof let x be set; assume x in C; then x = A or x = B by TARSKI:def 2; hence thesis; end; hence thesis by SETFAM_1:def 7; end; theorem Th8: for A,B,C being Subset of X holds {A,B,C} is non empty Subset-Family of X proof let A,B,C be Subset of X; set D = {A,B,C}; D c= bool X proof let x be set; assume x in D; then x = A or x = B or x = C by ENUMSET1:def 1; hence thesis; end; hence thesis by ENUMSET1:def 1,SETFAM_1:def 7; end; theorem Th9: {{}} is Subset-Family of X proof {} c= X by XBOOLE_1:2; then {{}} c= bool X by ZFMISC_1:37; hence thesis by SETFAM_1:def 7; end; scheme DomsetFamEx {A() -> set,P[set]}: ex F being non empty Subset-Family of A() st for B being set holds B in F iff (B c= A() & P[B]) provided A1: ex B being set st B c= A() & P[B] proof defpred X[set] means ex Z being set st $1 = Z & P[Z]; consider X being set such that A2:for x being set holds x in X iff x in bool A() & X[x] from Separation; consider B being set such that A3: B c= A() & P[B] by A1; X c= bool A() proof let x be set; assume x in X; hence thesis by A2; end; then reconsider X as non empty Subset-Family of A() by A2,A3,SETFAM_1:def 7; take X; for B being set holds B in X iff (B c= A() & P[B]) proof let B be set; thus B in X implies B c= A() & P[B] proof assume A4:B in X; then ex Z being set st B = Z & P[Z] by A2; hence thesis by A4; end; assume B c= A() & P[B]; hence B in X by A2; end; hence thesis; end; definition let X be set; let S be non empty Subset-Family of X; canceled; func X\S -> Subset-Family of X means :Def2: for A being set holds A in it iff ex B being set st B in S & A = X \ B; existence proof defpred P[set] means ex B being set st B in S & $1 = X \ B; A1:ex C being set st C c= X & P[C] proof consider C0 being Element of S; take X \ C0; thus thesis by XBOOLE_1:36; end; ex R being non empty Subset-Family of X st for A being set holds A in R iff A c= X & P[A] from DomsetFamEx(A1); then consider R being Subset-Family of X such that A2:for A being set holds A in R iff (A c= X & (ex B being set st B in S & A = X \ B)); take R; let A be set; thus A in R implies (ex B being set st B in S & A = X \ B) by A2; given B being set such that A3: B in S & A = X \ B; A c= X by A3,XBOOLE_1:36; hence A in R by A2,A3; end; uniqueness proof let R1,R2 be Subset-Family of X such that A4:for A being set holds A in R1 iff (ex B being set st B in S & A = X \ B) and A5:for A being set holds A in R2 iff (ex B being set st B in S & A = X \ B); R1 = R2 proof for A being set holds A in R1 iff A in R2 proof let A be set; hereby assume A in R1; then ex B being set st B in S & A = X \ B by A4; hence A in R2 by A5; end; assume A in R2; then ex B being set st B in S & A = X \ B by A5; hence thesis by A4; end; hence thesis by TARSKI:2; end; hence thesis; end; end; definition let X be set; let S be non empty Subset-Family of X; cluster X\S -> non empty; coherence proof consider x being set such that A1: x in S by XBOOLE_0:def 1; X\x in X\S by A1,Def2; hence thesis; end; end; canceled 4; theorem Th14: for S being non empty Subset-Family of X holds S = X \ (X \ S) proof let S be non empty Subset-Family of X; A1:S c= X \ (X \ S) proof let A be set; assume A2:A in S; then A3:X\A in X \ S by Def2; X \ (X \ A) = X /\ A by XBOOLE_1:48; then A = X \ (X\A) by A2,XBOOLE_1:28; hence thesis by A3,Def2; end; X \ (X \ S) c= S proof let A be set; assume A in X \ (X \ S); then consider B being set such that A4:B in X \ S & A = X \ B by Def2; consider C being set such that A5:C in S & B = X \ C by A4,Def2; A = X /\ C by A4,A5,XBOOLE_1:48; hence thesis by A5,XBOOLE_1:28; end; hence thesis by A1,XBOOLE_0:def 10; end; theorem Th15: for S being non empty Subset-Family of X holds meet S = X \ union (X \ S) & union S = X \ meet (X \ S) proof let S be non empty Subset-Family of X; thus meet S = X \ union (X \ S) proof thus meet S c= X \ union (X \ S) proof let x be set; assume A1: x in meet S; not x in union (X \ S) proof assume x in union (X \ S); then consider Z being set such that A2:x in Z & Z in X \ S by TARSKI:def 4; consider B being set such that A3:B in S & Z = X \ B by A2,Def2; not x in B by A2,A3,XBOOLE_0:def 4; hence thesis by A1,A3,SETFAM_1:def 1; end; hence thesis by A1,XBOOLE_0:def 4; end; thus X \ union (X \ S) c= meet S proof let x be set; assume x in X \ union (X \ S); then A4:x in X & not x in union (X \ S) by XBOOLE_0:def 4; for Y being set st Y in S holds x in Y proof let Y be set; assume that A5:Y in S and A6:not x in Y; A7:x in X\Y by A4,A6,XBOOLE_0:def 4; X\Y in X \ S by A5,Def2; hence thesis by A4,A7,TARSKI:def 4; end; hence thesis by SETFAM_1:def 1; end; end; thus union S c= X \ meet (X \ S) proof let x be set; assume x in union S; then consider Y being set such that A8:x in Y & Y in S by TARSKI:def 4; not x in meet (X \ S) proof assume A9:x in meet (X \ S); set Z = X \ Y; A10:Z in X \ S by A8,Def2; not x in Z by A8,XBOOLE_0:def 4; hence thesis by A9,A10,SETFAM_1:def 1; end; hence thesis by A8,XBOOLE_0:def 4; end; let x be set; assume x in X \ meet (X \ S); then A11:x in X & not x in meet (X \ S) by XBOOLE_0:def 4; then consider Z being set such that A12:Z in X \ S & not x in Z by SETFAM_1:def 1; ex Y being set st Y in S & x in Y proof set Y = X \ Z; Y in X \ (X \ S) by A12,Def2; then A13:Y in S by Th14; x in Y by A11,A12,XBOOLE_0:def 4; hence thesis by A13; end; hence thesis by TARSKI:def 4; end; :: :: Field Subset of X and nonnegative measure :: definition let X be set; let IT be Subset-Family of X; redefine attr IT is compl-closed means :Def3: for A being set holds A in IT implies X\A in IT; compatibility proof hereby assume A1: IT is compl-closed; let A be set; assume A2: A in IT; then reconsider A' = A as Subset of X; A'` in IT by A1,A2,PROB_1:def 1; hence X\A in IT by SUBSET_1:def 5; end; assume A3: for A being set holds A in IT implies X\A in IT; let A be Subset of X; assume A in IT; then X\A in IT by A3; hence thesis by SUBSET_1:def 5; end; end; theorem Th16: for X being set, F being Subset-Family of X st F is cup-closed compl-closed holds F is cap-closed proof let X be set; let S be Subset-Family of X; assume A1: S is cup-closed compl-closed; let A,B be set; assume A2:A in S & B in S; then X \ A in S & X \ B in S by A1,Def3; then A3:(X \ A) \/ (X \ B) in S by A1,FINSUB_1:def 1; A /\ B c= A by XBOOLE_1:17; then A /\ B c= X by A2,XBOOLE_1:1; then 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; hence thesis by A1,A3,Def3; end; definition let X be set; cluster cup-closed compl-closed -> cap-closed Subset-Family of X; coherence by Th16; cluster cap-closed compl-closed -> cup-closed Subset-Family of X; coherence proof let S be Subset-Family of X; assume A1: S is cap-closed compl-closed; let A,B be set; assume A2:A in S & B in S; then X \ A in S & X \ B in S by A1,Def3; then A3:(X \ A) /\ (X \ B) in S by A1,FINSUB_1:def 2; A \/ B c= X by A2,XBOOLE_1:8; then A \/ B = X /\ (A \/ B) by XBOOLE_1:28 .= X \ (X \ (A \/ B)) by XBOOLE_1:48 .= X \ ((X \ A) /\ (X \ B)) by XBOOLE_1:53; hence thesis by A1,A3,Def3; end; end; theorem Th17: for S being Field_Subset of X holds S = X \ S proof let S be Field_Subset of X; for A being set holds A in S iff A in X \ S proof let A be set; hereby assume A1:A in S; then A2:X \ A in S by Def3; A = X \ (X \ A) proof for x be set holds x in A iff x in X\(X\A) proof let x be set; hereby assume x in A; then x in X & not x in X\A by A1,XBOOLE_0:def 4; hence x in X\(X\A) by XBOOLE_0:def 4; end; assume x in X\(X\A); then x in X & not x in X\A by XBOOLE_0:def 4; hence thesis by XBOOLE_0:def 4; end; hence thesis by TARSKI:2; end; hence A in X \ S by A2,Def2; end; assume A in X \ S; then ex B being set st B in S & A = X \ B by Def2; hence thesis by Def3; end; hence thesis by TARSKI:2; end; theorem for M being set holds M is Field_Subset of X iff ex S being non empty Subset-Family of X st M = S & for A being set st A in S holds X\A in S & for A,B being set st A in S & B in S holds A \/ B in S proof let M be set; hereby assume A1: M is Field_Subset of X; then reconsider S = M as non empty Subset-Family of X; take S; thus M = S; thus for A being set st A in S holds X\A in S & for A,B being set st (A in S & B in S) holds A \/ B in S by A1,Def3,FINSUB_1:def 1; end; given S being non empty Subset-Family of X such that A2: M = S & for A being set st A in S holds X\A in S & for A,B being set st A in S & B in S holds A \/ B in S; A3:S is cup-closed proof let A, B be set; assume A in S & B in S; hence A \/ B in S by A2; end; S is compl-closed proof let A be set; assume A in S; hence X\A in S by A2; end; hence thesis by A2,A3,Th16; end; theorem Th19: for S being non empty Subset-Family of X holds S is Field_Subset of X iff for A being set st A in S holds X\A in S & for A,B being set st A in S & B in S holds A /\ B in S proof let S be non empty Subset-Family of X; hereby assume A1: S is Field_Subset of X; for A,B being set holds A in S & B in S implies A /\ B in S proof let A,B be set; assume A2:A in S & B in S; then X \ A in S & X \ B in S by A1,Def3; then A3:(X \ A) \/ (X \ B) in S by A1,FINSUB_1:def 1; A /\ B c= A by XBOOLE_1:17; then A /\ B c= X by A2,XBOOLE_1:1; then 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; hence thesis by A1,A3,Def3; end; hence for A being set st A in S holds X\A in S & for A,B being set st A in S & B in S holds A /\ B in S by A1,Def3; end; assume A4: for A being set st A in S holds X\A in S & for A,B being set st A in S & B in S holds A /\ B in S; then A5:for A being set st A in S holds X\A in S; for A,B being set st A in S & B in S holds A \/ B in S proof let A,B be set; assume A6:A in S & B in S; then X \ A in S & X \ B in S by A4; then A7:(X \ A) /\ (X \ B) in S by A4; A \/ B c= X by A6,XBOOLE_1:8; then A \/ B = X /\ (A \/ B) by XBOOLE_1:28 .= X \ (X \ (A \/ B)) by XBOOLE_1:48 .= X \ ((X \ A) /\ (X \ B)) by XBOOLE_1:53; hence thesis by A4,A7; end; then S is cup-closed compl-closed by A5,Def3,FINSUB_1:def 1; hence thesis by Th16; end; theorem Th20: for S being Field_Subset of X holds for A,B being set st A in S & B in S holds A \ B in S proof let S be Field_Subset of X; let A,B be set; assume A1:A in S & B in S; then A2:X \ B in S by Th19; A /\ (X \ B) = (A /\ X) \ B by XBOOLE_1:49 .= A \ B by A1,XBOOLE_1:28; hence thesis by A1,A2,Th19; end; theorem for S being Field_Subset of X holds {} in S & X in S by PROB_1:10,11; definition let S be non empty set; let F be Function of S,ExtREAL; let A be Element of S; redefine func F.A -> R_eal; coherence by FUNCT_2:7; end; definition let X be non empty set, F be Function of X,ExtREAL; redefine attr F is nonnegative means :Def4:for A being Element of X holds 0. <=' F.A; compatibility by SUPINF_2:58; end; canceled; theorem Th23: for S being Field_Subset of X holds ex M being Function of S,ExtREAL st M is nonnegative & M.{} = 0.& for A,B being Element of S st A misses B holds M.(A \/ B) = M.A + M.B proof let S be Field_Subset of X; deffunc F(Element of S) = 0.; ex f being Function of S,ExtREAL st for x being Element of S holds f.x = F(x) from LambdaD; then consider M being Function of S,ExtREAL such that A1:for x being Element of S holds M.x = 0.; A2:M is nonnegative proof for x being Element of S holds 0.<=' M.x by A1; hence thesis by Def4; end; A3:M.{} = 0. proof reconsider A = {} as Element of S by PROB_1:10; M.A = 0.by A1; hence thesis; end; A4:for A,B being Element of S st A misses B holds M.(A \/ B) = M.A + M.B proof let A,B be Element of S; assume A misses B; A5:M.A = 0.& M.B = 0.by A1; reconsider A,B as set; A6: A \/ B is Element of S by FINSUB_1:def 1; reconsider A,B as Element of S; 0.= M.A + M.B by A5,SUPINF_2:18; hence thesis by A1,A6; end; take M; thus thesis by A2,A3,A4; end; definition let X be set, S be Field_Subset of X; mode Measure of S -> Function of S,ExtREAL means :Def5:it is nonnegative & it.{} = 0. & for A,B being Element of S st A misses B holds it.(A \/ B) = it.A + it.B; existence by Th23; end; canceled; theorem Th25: for S being Field_Subset of X, M being Measure of S, A,B being Element of S holds A c= B implies M.A <=' M.B proof let S be Field_Subset of X, M be Measure of S, A,B be Element of S; assume A1:A c= B; reconsider C = B \ A as Element of S by Th20; A misses C by XBOOLE_1:79; then M.(A \/ C) = M.A + M.C by Def5; then A2:M.B = M.A + M.C by A1,XBOOLE_1:45; M is nonnegative by Def5; then 0.<=' M.A & 0.<=' M.C by Def4; hence thesis by A2,SUPINF_2:20; end; theorem Th26: for S being Field_Subset of X, M being Measure of S, A,B being Element of S holds A c= B & M.A <' +infty implies M.(B \ A) = M.B - M.A proof let S be Field_Subset of X, M be Measure of S, A,B be Element of S; assume A1:A c= B & M.A <' +infty; reconsider C = B \ A as Element of S by Th20; A misses C by XBOOLE_1:79; then M.(A \/ C) = M.A + M.C by Def5; then A2:M.B = M.A + M.C by A1,XBOOLE_1:45; M is nonnegative by Def5; then 0.<=' M.A & 0.<=' M.C & M.A <' +infty by A1,Def4; hence thesis by A2,Th5; end; definition let X be set; cluster non empty compl-closed cap-closed Subset-Family of X; existence proof consider S being non empty compl-closed cap-closed Subset-Family of X; take S; thus thesis; end; end; definition let X be set, S be non empty cup-closed Subset-Family of X, A,B be Element of S; redefine func A \/ B -> Element of S; coherence by FINSUB_1:def 1; end; definition let X be set, S be Field_Subset of X, A,B be Element of S; redefine func A /\ B -> Element of S; coherence by Th19; redefine func A \ B -> Element of S; coherence by Th20; end; theorem Th27: for S being Field_Subset of X, M being Measure of S, A,B being Element of S holds M.(A \/ B) <=' M.A + M.B proof let S be Field_Subset of X, M be Measure of S, A,B be Element of S; set C = B \ A; A1:A misses C by XBOOLE_1:79; A2:M.(A \/ B) = M.(A \/ C) by XBOOLE_1:39 .= M.A + M.C by A1,Def5; A3:C c= B by XBOOLE_1:36; M is nonnegative by Def5; then 0.<=' M.A & 0.<=' M.C & M.A <=' M.A & M.C <=' M.B by A3,Def4,Th25; hence thesis by A2,Th4; end; definition let X be set, S be Field_Subset of X, M be Measure of S, A be set; pred A is_measurable M means :Def6:A in S; end; canceled; theorem for S being Field_Subset of X, M being Measure of S holds {} is_measurable M & X is_measurable M & for A,B being set st A is_measurable M & B is_measurable M holds X \ A is_measurable M & A \/ B is_measurable M & A /\ B is_measurable M proof let S be Field_Subset of X, M be Measure of S; A1:{} in S & X in S by PROB_1:10,11; for A,B being set st A is_measurable M & B is_measurable M holds X \ A is_measurable M & A \/ B is_measurable M & A /\ B is_measurable M proof let A,B be set; assume A is_measurable M & B is_measurable M; then A in S & B in S by Def6; then X \ A in S & A \/ B in S & A /\ B in S & A \ B in S by Th19,Th20,FINSUB_1:def 1; hence thesis by Def6; end; hence thesis by A1,Def6; end; definition let X be set, S be Field_Subset of X, M be Measure of S; mode measure_zero of M -> Element of S means :Def7:M.it = 0.; existence proof reconsider A = {} as Element of S by PROB_1:10; take A; thus thesis by Def5; end; end; canceled; theorem for S being Field_Subset of X, M being Measure of S, A being Element of S, B being measure_zero of M st A c= B holds A is measure_zero of M proof let S be Field_Subset of X, M be Measure of S, A be Element of S, B be measure_zero of M; assume A1:A c= B; M is nonnegative by Def5; then 0.<=' M.A & M.A <=' M.B by A1,Def4,Th25; then 0.<=' M.A & M.A <=' 0.by Def7; then M.A = 0.by SUPINF_1:22; hence thesis by Def7; end; theorem for S being Field_Subset of X, M being Measure of S, A,B being measure_zero of M holds A \/ B is measure_zero of M & A /\ B is measure_zero of M & A \ B is measure_zero of M proof let S be Field_Subset of X, M be Measure of S, A,B be measure_zero of M; A1:M.A = 0.& M.B = 0.by Def7; M is nonnegative by Def5; then A2: 0.<=' M.(A \/ B) & 0.<=' M.(A /\ B) & 0.<=' M.(A \ B) by Def4; M.(A \/ B) <=' 0.+ 0.by A1,Th27; then M.(A \/ B) <=' 0.by SUPINF_2:18; then A3:M.(A \/ B) = 0.by A2,SUPINF_1:22; A /\ B c= A by XBOOLE_1:17; then M.(A /\ B) <=' 0.by A1,Th25; then A4:M.(A /\ B) = 0.by A2,SUPINF_1:22; A \ B c= A by XBOOLE_1:36; then M.(A \ B) <=' 0.by A1,Th25; then M.(A \ B) = 0.by A2,SUPINF_1:22; hence thesis by A3,A4,Def7; end; theorem for S being Field_Subset of X, M being Measure of S, A being Element of S, B being measure_zero of M holds M.(A \/ B) = M.A & M.(A /\ B) = 0.& M.(A \ B) = M.A proof let S be Field_Subset of X, M be Measure of S, A be Element of S, B be measure_zero of M; A1:M.B = 0.by Def7; M is nonnegative by Def5; then A2: 0.<=' M.(A \/ B) & 0.<=' M.(A /\ B) & 0.<=' M.(A \ B) by Def4; M.(A \/ B) <=' M.A + 0.by A1,Th27; then A3:M.(A \/ B) <=' M.A by SUPINF_2:18; A c= A \/ B by XBOOLE_1:7; then A4: M.A <=' M.(A \/ B) by Th25; A /\ B c= B by XBOOLE_1:17; then A5: M.(A /\ B) <=' 0.by A1,Th25; then A6:M.(A /\ B) = 0.by A2,SUPINF_1:22; A \ B c= A by XBOOLE_1:36; then A7:M.(A \ B) <=' M.A by Th25; M.A = M.((A /\ B) \/ (A \ B)) by XBOOLE_1:51; then M.A <=' 0.+ M.(A \ B) by A6,Th27; then M.A <=' M.(A \ B) by SUPINF_2:18; hence thesis by A2,A3,A4,A5,A7,SUPINF_1:22; end; :: :: sigma_Field Subset of X and sigma_additive nonnegative measure :: theorem Th34: for A being Subset of X ex F being Function of NAT,bool X st rng F = {A} proof let A be Subset of X; reconsider F = NAT --> A as Function of NAT, bool X by FUNCOP_1:57; take F; thus thesis by FUNCOP_1:14; end; theorem Th35: for A being set ex F being Function of NAT,{A} st for n being Nat holds F.n = A proof let A be set; A in {A} by TARSKI:def 1; then reconsider F = NAT --> A as Function of NAT, {A} by FUNCOP_1:57; take F; thus thesis by TARSKI:def 1; end; definition let X be set; cluster non empty countable Subset-Family of X; existence proof A1:{} is Subset of X by XBOOLE_1:2; reconsider S = {{}} as Subset-Family of X by Th9; take S; thus S is non empty; thus S is empty or ex F being Function of NAT,bool X st S = rng F by A1,Th34; end; end; definition let X be set; mode N_Sub_set_fam of X is non empty countable Subset-Family of X; end; canceled 2; theorem Th38: for A,B,C being Subset of X ex F being Function of NAT,bool X st rng F = {A,B,C} & F.0 = A & F.1 = B & for n being Nat st 1 < n holds F.n = C proof let A,B,C be Subset of X; defpred P0[Nat,set] means ($1 = 0 implies $2 = A) & ($1 = 1 implies $2 = B) & (1 < $1 implies $2 = C); defpred P[set] means ex n being Nat st ex X being set st ($1 = [n,X] & P0[n,X]); ex GRAF being set st for x being set holds x in GRAF iff x in [:NAT,bool X:] & P[x] from Separation; then consider GRAF being set such that A1:for x being set holds x in GRAF iff x in [:NAT,bool X:] & P[x]; A2:for x being set holds x in GRAF iff (x = [0,A] or x = [1,B] or (ex n being Nat st (x = [n,C] & 1 < n))) proof let x be set; A3:x in GRAF implies (x = [0,A] or x = [1,B] or (ex n being Nat st (x = [n,C] & 1 < n))) proof assume x in GRAF; then consider n being Nat such that A4:ex X0 being set st x = [n,X0] & P0[n,X0] by A1; n = 0 or n = 1 or 1 < n by CQC_THE1:2; hence thesis by A4; end; (x = [0,A] or x = [1,B] or (ex n being Nat st (x = [n,C] & 1 < n))) implies x in GRAF proof assume A5:x = [0,A] or x = [1,B] or (ex n being Nat st (x = [n,C] & 1 < n)); A6:x = [0,A] implies x in GRAF proof assume x = [0,A]; then x in [:NAT,bool X:] & P[x] by ZFMISC_1:106; hence thesis by A1; end; A7:x = [1,B] implies x in GRAF proof assume x = [1,B]; then x in [:NAT,bool X:] & P[x] by ZFMISC_1:106; hence thesis by A1; end; (ex n being Nat st (x = [n,C] & 1 < n)) implies x in GRAF proof given n being Nat such that A8: x = [n,C] & 1 < n; P0[n,C] by A8; then x in [:NAT,bool X:] & P[x] by A8,ZFMISC_1:106; hence thesis by A1; end; hence thesis by A5,A6,A7; end; hence thesis by A3; end; A9:for p being set st p in GRAF ex x,y being set st [x,y] = p proof let p be set; assume p in GRAF; then p = [0,A] or p = [1,B] or (ex n being Nat st p = [n,C] & 1 < n) by A2; hence thesis; end; for x,y1,y2 being set st [x,y1] in GRAF & [x,y2] in GRAF holds y1 = y2 proof let x,y1,y2 be set; assume A10:[x,y1] in GRAF & [x,y2] in GRAF; then A11:[x,y1] in [:NAT,bool X:] & P[[x,y1]] by A1; A12:[x,y2] in [:NAT,bool X:] & P[[x,y2]] by A1,A10; then reconsider x as Nat by ZFMISC_1:106; per cases by CQC_THE1:2; suppose A13: x = 0; then y1 = A by A11,ZFMISC_1:33; hence thesis by A12,A13,ZFMISC_1:33; suppose A14: x = 1; then y1 = B by A11,ZFMISC_1:33; hence thesis by A12,A14,ZFMISC_1:33; suppose A15: x > 1; then y1 = C by A11,ZFMISC_1:33; hence thesis by A12,A15,ZFMISC_1:33; end; then reconsider F = GRAF as Function by A9,FUNCT_1:2; A16:dom F = NAT proof for x being set holds x in NAT iff ex y being set st [x,y] in F proof let x be set; thus x in NAT implies ex y being set st [x,y] in F proof assume x in NAT; then reconsider x as Nat; A17:x = 0 implies ex y being set st [x,y] in F proof assume A18:x = 0; take A; thus thesis by A2,A18; end; A19:x = 1 implies ex y being set st [x,y] in F proof assume A20:x = 1; take B; thus thesis by A2,A20; end; 1 < x implies ex y being set st [x,y] in F proof assume A21:1 < x; take C; thus thesis by A2,A21; end; hence thesis by A17,A19,CQC_THE1:2; end; given y being set such that A22:[x,y] in F; [x,y] in [:NAT,bool X:] by A1,A22; hence thesis by ZFMISC_1:106; end; hence thesis by RELAT_1:def 4; end; A23:rng F = {A,B,C} proof for y being set holds y in {A,B,C} iff ex x being set st x in dom F & y = F.x proof let y be set; thus y in {A,B,C} implies ex x being set st (x in dom F & y = F.x) proof assume A24: y in {A,B,C}; per cases by A24,ENUMSET1:def 1; suppose y = A; then A25:[0,y] in F by A2; take 0; thus thesis by A25,FUNCT_1:8; suppose y = B; then A26:[1,y] in F by A2; take 1; thus thesis by A26,FUNCT_1:8; suppose y = C; then A27:[2,y] in F by A2; take 2; thus thesis by A27,FUNCT_1:8; end; given x being set such that A28:x in dom F & y = F.x; reconsider x as Nat by A16,A28; per cases by CQC_THE1:2; suppose A29:x = 0; [0,A] in F by A2; then A = F.x by A29,FUNCT_1:8; hence thesis by A28,ENUMSET1:def 1; suppose A30:x = 1; [1,B] in F by A2; then B = F.x by A30,FUNCT_1:8; hence thesis by A28,ENUMSET1:def 1; suppose 1 < x; then [x,C] in F by A2; then y = C by A28,FUNCT_1:8; hence thesis by ENUMSET1:def 1; end; hence thesis by FUNCT_1:def 5; end; rng F c= bool X proof let a be set; assume a in rng F; then a = A or a = B or a = C by A23,ENUMSET1:def 1; hence thesis; end; then reconsider F as Function of NAT,bool X by A16,FUNCT_2:4; take F; F.0 = A & F.1 = B & for n being Nat st 1 < n holds F.n = C proof A31: [0,A] in F by A2; A32: [1,B] in F by A2; for n being Nat st 1 < n holds F.n = C proof let n be Nat; assume 1 < n; then [n,C] in F by A2; hence thesis by FUNCT_1:8; end; hence thesis by A31,A32,FUNCT_1:8; end; hence thesis by A23; end; theorem for A,B being Subset of X holds {A,B,{}} is N_Sub_set_fam of X proof let A,B be Subset of X; {} is Subset of X by XBOOLE_1:2; then consider C being Subset of X such that A1:C = {}; ex F being Function of NAT,bool X st rng F = {A,B,C} & F.0 = A & F.1 = B & for n being Nat st 1 < n holds F.n = C by Th38; hence thesis by A1,Th8,SUPINF_2:def 14; end; theorem Th40: for A,B being Subset of X ex F being Function of NAT,bool X st rng F = {A,B} & F.0 = A & for n being Nat st 0 < n holds F.n = B proof let A,B be Subset of X; consider F being Function of NAT,bool X such that A1:rng F = {A,B,B} & F.0 = A & F.1 = B & for n being Nat st 1 < n holds F.n = B by Th38; A2:{A,B,B} = {B,B,A} by ENUMSET1:100 .= {A,B} by ENUMSET1:70; A3:for n being Nat st 0 < n holds F.n = B proof let n be Nat; assume 0 < n; then n = 1 or 1 < n by CQC_THE1:2; hence thesis by A1; end; take F; thus thesis by A1,A2,A3; end; theorem Th41: for A,B being Subset of X holds {A,B} is N_Sub_set_fam of X proof let A,B be Subset of X; ex F being Function of NAT,bool X st (rng F = {A,B} & F.0 = A & for n being Nat st 0 < n holds F.n = B) by Th40; hence thesis by Th7,SUPINF_2:def 14; end; theorem Th42: for S being N_Sub_set_fam of X holds X \ S is N_Sub_set_fam of X proof let S be N_Sub_set_fam of X; consider F being Function of NAT,bool X such that A1:S = rng F by SUPINF_2:def 14; A2:NAT = dom F by FUNCT_2:def 1; A3:for n being set st n in NAT ex Y being set st Y in S & Y=F.n proof let n be set; assume A4:n in NAT; take F.n; thus thesis by A1,A4,FUNCT_2:6; end; defpred P[set,set] means ex V being set st V = F.$1 & $2 = X \ V; A5:for n being set st n in NAT ex y being set st y in X\S & P[n,y] proof let n be set; assume n in NAT; then consider V being set such that A6: V in S & V=F.n by A3; take X \ V; thus thesis by A6,Def2; end; A7:ex G being Function of NAT,X \ S st for n being set st n in NAT holds P[n,G.n] from FuncEx1(A5); ex G being Function of NAT,bool X st X \ S = rng G proof consider G being Function of NAT,X \ S such that A8:for n being set st n in NAT holds (ex V being set st V = F.n & G.n = X \ V) by A7; A9:dom G = NAT & rng G c= X \ S by FUNCT_2:def 1,RELSET_1:12; A10: X \ S c= rng G proof let x be set; assume x in X \ S; then consider B being set such that A11:B in S & x = X \ B by Def2; consider n being set such that A12:n in NAT & F.n = B by A1,A2,A11,FUNCT_1:def 5; ex V being set st V = F.n & G.n = X \ V by A8,A12; hence thesis by A9,A11,A12,FUNCT_1:def 5; end; reconsider G as Function of NAT,bool X by FUNCT_2:9; take G; thus thesis by A9,A10,XBOOLE_0:def 10; end; hence thesis by SUPINF_2:def 14; end; definition let X be set; let IT be Subset-Family of X; canceled; attr IT is sigma_Field_Subset-like means :Def9: for M being N_Sub_set_fam of X st M c= IT holds union M in IT; end; definition let X be set; cluster non empty compl-closed sigma_Field_Subset-like Subset-Family of X; existence proof reconsider S = {{},X} as non empty Subset-Family of X by PROB_1:14; take S; thus S is non empty; thus for A being set holds A in S implies X\A in S proof let A be set; assume A1: A in S; A2: A = {} implies X\A in S by TARSKI:def 2; A = X implies X\A in S proof assume A = X; then X\A = {} by XBOOLE_1:37; hence thesis by TARSKI:def 2; end; hence thesis by A1,A2,TARSKI:def 2; end; let M be N_Sub_set_fam of X; assume A3:M c= S; A4:X in M implies union M in S proof assume X in M; then X c= union M by ZFMISC_1:92; then X = union M by XBOOLE_0:def 10; hence thesis by TARSKI:def 2; end; not X in M implies union M in S proof assume not X in M; then for A being set st A in M holds A c= {} by A3,TARSKI:def 2; then union M c= {} by ZFMISC_1:94; then union M = {} by XBOOLE_1:3; hence thesis by TARSKI:def 2; end; hence thesis by A4; end; end; definition let X be set; mode sigma_Field_Subset of X is compl-closed sigma_Field_Subset-like (non empty Subset-Family of X); end; Lm1: for S being non empty Subset-Family of X st S is sigma_Field_Subset of X holds S is Field_Subset of X proof let S be non empty Subset-Family of X; assume A1:S is sigma_Field_Subset of X; for A,B being set st A in S & B in S holds A \/ B in S proof let A,B be set; assume A2:A in S & B in S; then reconsider A,B as Subset of X; A3:{A,B} is N_Sub_set_fam of X by Th41; A4:union{A,B} = A \/ B by ZFMISC_1:93; {A,B} c= S proof for x being set holds x in {A,B} implies x in S by A2,TARSKI:def 2; hence thesis by TARSKI:def 3; end; hence thesis by A1,A3,A4,Def9; end; then S is cup-closed by FINSUB_1:def 1; hence thesis by A1,Th16; end; canceled 2; theorem Th45: for S being sigma_Field_Subset of X holds {} in S & X in S proof let S be sigma_Field_Subset of X; S is Field_Subset of X by Lm1; hence thesis by PROB_1:10,11; end; definition let X be set; cluster -> non empty sigma_Field_Subset of X; coherence; end; theorem Th46: for S being sigma_Field_Subset of X, A,B being set st A in S & B in S holds A \/ B in S & A /\ B in S proof let S be sigma_Field_Subset of X; let A,B be set; assume A1:A in S & B in S; S is Field_Subset of X by Lm1; hence thesis by A1,Th19,FINSUB_1:def 1; end; theorem Th47: for S being sigma_Field_Subset of X, A,B being set st A in S & B in S holds A \ B in S proof let S be sigma_Field_Subset of X; let A,B be set; assume A1:A in S & B in S; S is Field_Subset of X by Lm1; hence thesis by A1,Th20; end; theorem for S being sigma_Field_Subset of X holds S = X \ S proof let S be sigma_Field_Subset of X; S is Field_Subset of X by Lm1; hence thesis by Th17; end; theorem Th49: for S being non empty Subset-Family of X holds ((for A being set holds A in S implies X\A in S) & (for M being N_Sub_set_fam of X st M c= S holds meet M in S)) iff S is sigma_Field_Subset of X proof let S be non empty Subset-Family of X; hereby assume A1: (for A being set holds A in S implies X\A in S) & (for M being N_Sub_set_fam of X st M c= S holds meet M in S); for M being N_Sub_set_fam of X st M c= S holds union M in S proof let M be N_Sub_set_fam of X; assume A2: M c= S; A3: X \ M is N_Sub_set_fam of X by Th42; X \ M c= S proof let y be set; assume y in X \ M; then consider B being set such that A4:B in M & y = X \ B by Def2; thus thesis by A1,A2,A4; end; then meet (X \ M) in S by A1,A3; then X \ meet (X \ M) in S by A1; hence thesis by Th15; end; hence S is sigma_Field_Subset of X by A1,Def3,Def9; end; assume A5: S is sigma_Field_Subset of X; for M being N_Sub_set_fam of X st M c= S holds meet M in S proof let M be N_Sub_set_fam of X; assume A6: M c= S; A7: X \ M is N_Sub_set_fam of X by Th42; X \ M c= S proof let y be set; assume y in X \ M; then consider B being set such that A8:B in M & y = X \ B by Def2; thus thesis by A5,A6,A8,Def3; end; then union (X \ M) in S by A5,A7,Def9; then X \ union (X \ M) in S by A5,Def3; hence thesis by Th15; end; hence thesis by A5,Def3; end; definition let X be set; let S be sigma_Field_Subset of X; cluster disjoint_valued Function of NAT, S; existence proof consider F being Function of NAT,{{}} such that A1:for n being Nat holds F.n = {} by Th35; {} in S by Th45; then {{}} c= S by ZFMISC_1:37; then reconsider F as Function of NAT,S by FUNCT_2:9; take F; A2:for n being set holds F.n = {} proof let n be set; per cases; suppose n in dom F; then n in NAT by FUNCT_2:def 1; hence thesis by A1; suppose not n in dom F; hence thesis by FUNCT_1:def 4; end; thus for x,y being set st x <> y holds F.x misses F.y proof let x,y be set; F.x = {} & F.y = {} by A2; hence thesis by XBOOLE_1:65; end; end; end; definition let X be set; let S be sigma_Field_Subset of X; mode Sep_Sequence of S is disjoint_valued Function of NAT, S; end; definition let X be set; let S be sigma_Field_Subset of X; let F be Function of NAT,S; redefine func rng F -> non empty Subset-Family of X; coherence proof 0 in NAT; then rng F <> {} by FUNCT_2:6; then consider x being set such that A1: x in rng F by XBOOLE_0:def 1; rng F c= S by RELSET_1:12; then rng F c= bool X by XBOOLE_1:1; hence thesis by A1,SETFAM_1:def 7; end; end; canceled 2; theorem Th52: for S being sigma_Field_Subset of X, F being Function of NAT,S holds rng F is N_Sub_set_fam of X proof let S be sigma_Field_Subset of X; let F be Function of NAT,S; ex H being Function of NAT,bool X st rng F = rng H proof rng F c= bool X; then reconsider F as Function of NAT,bool X by FUNCT_2:8; take F; thus thesis; end; hence thesis by SUPINF_2:def 14; end; theorem Th53: for S being sigma_Field_Subset of X, F being Function of NAT,S holds union rng F is Element of S proof let S be sigma_Field_Subset of X, F be Function of NAT,S; A1:rng F is N_Sub_set_fam of X by Th52; rng F c= S by RELSET_1:12; hence thesis by A1,Def9; end; theorem Th54: for Y,S being non empty set, F being Function of Y,S, M being Function of S,ExtREAL st M is nonnegative holds M*F is nonnegative proof let Y,S be non empty set; let F be Function of Y,S; let M be Function of S,ExtREAL; assume A1:M is nonnegative; for n being Element of Y holds 0.<=' (M*F).n proof let n be Element of Y; dom (M*F) = Y & rng (M*F) c= ExtREAL by FUNCT_2:def 1,RELSET_1:12; then (M*F).n = M.(F.n) by FUNCT_1:22; hence thesis by A1,Def4; end; hence thesis by SUPINF_2:58; end; theorem Th55: for S being sigma_Field_Subset of X, a,b being R_eal holds ex M being Function of S,ExtREAL st for A being Element of S holds (A = {} implies M.A = a) & (A <> {} implies M.A = b) proof let S be sigma_Field_Subset of X, 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 S 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 S,ExtREAL st for x being set st x in S holds P[x,F.x] from FuncEx1(A1); then consider M being Function of S,ExtREAL such that A2:for x being set st x in S holds P[x,M.x]; take M; thus thesis by A2; end; theorem for S being sigma_Field_Subset of X ex M being Function of S,ExtREAL st for A being Element of S holds (A = {} implies M.A = 0.) & (A <> {} implies M.A = +infty) by Th55; theorem Th57: for S being sigma_Field_Subset of X ex M being Function of S,ExtREAL st for A being Element of S holds M.A = 0. proof let S be sigma_Field_Subset of X; consider M being Function of S,ExtREAL such that A1:for A being Element of S holds (A = {} implies M.A = 0.) & (A <> {} implies M.A = 0.) by Th55; A2:for A being Element of S holds M.A = 0. proof let A be Element of S; A = {} implies M.A = 0.by A1; hence thesis by A1; end; take M; thus thesis by A2; end; theorem Th58: for S being sigma_Field_Subset of X ex M being Function of S,ExtREAL st M is nonnegative & M.{} = 0. & for F being Sep_Sequence of S holds SUM(M*F) = M.(union rng F) proof let S be sigma_Field_Subset of X; consider M being Function of S,ExtREAL such that A1:for A being Element of S holds M.A = 0. by Th57; take M; for A being Element of S holds 0.<=' M.A by A1; hence A2: M is nonnegative by Def4; {} is Element of S by Th45; hence M.{} = 0. by A1; let F be Sep_Sequence of S; union rng F is Element of S by Th53; then A3:M.(union rng F) = 0.by A1; A4:M*F is nonnegative by A2,Th54; A5: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 A6:SUM(M*F) = Ser(M*F).0 by A4,SUPINF_2:67; Ser(M*F).0 = (M*F).0 by SUPINF_2:63; hence thesis by A3,A5,A6; end; definition let X be set; let S be sigma_Field_Subset of X; canceled; mode sigma_Measure of S -> Function of S,ExtREAL means :Def11:it is nonnegative & it.{} = 0.& for F being Sep_Sequence of S holds SUM(it*F) = it.(union rng F); existence by Th58; end; definition let X be set; cluster sigma_Field_Subset-like compl-closed -> cup-closed (non empty Subset-Family of X); coherence by Lm1; end; canceled; theorem Th60: for S being sigma_Field_Subset of X, M being sigma_Measure of S holds M is Measure of S proof let S be sigma_Field_Subset of X, M be sigma_Measure of S; A1:M is nonnegative & M.{} = 0.by Def11; for A,B being Element of S holds (A misses B implies M.(A \/ B) = M.A + M.B) proof let A,B be Element of S; assume A2:A misses B; A3:A in S & B in S & {} in S by Th45; A4: {} is Subset of X by XBOOLE_1:2; reconsider A,B as Subset of X; consider F being Function of NAT,bool X such that A5:rng F = {A,B,{}} & F.0 = A & F.1 = B & for n being Nat st 1 < n holds F.n = {} by A4,Th38; F is Function of NAT,S proof {A,B,{}} c= S proof for x being set holds x in {A,B,{}} implies x in S by A3,ENUMSET1:13; hence thesis by TARSKI:def 3; end; hence thesis by A5,FUNCT_2:8; end; then reconsider F as Function of NAT,S; A6: for n,m being Nat st n <> m holds F.n misses F.m proof let n,m be Nat; assume A7: n <> m; A8: (n = 0 or n = 1 or 1 < n) & (m = 0 or m = 1 or 1 < m) by CQC_THE1:2; A9:n = 0 & 1 < m implies F.n misses F.m proof assume n = 0 & 1 < m; then F.n = A & F.m = {} by A5; then F.n /\ F.m = {}; hence thesis by XBOOLE_0:def 7; end; A10:n = 1 & 1 < m implies F.n misses F.m proof assume n = 1 & 1 < m; then F.n = B & F.m = {} by A5; then F.n /\ F.m = {}; hence thesis by XBOOLE_0:def 7; end; A11:(1 < n & m = 0) implies F.n misses F.m proof assume 1 < n & m = 0; then F.n = {} & F.m = A by A5; then F.n /\ F.m = {}; hence thesis by XBOOLE_0:def 7; end; A12:1 < n & m = 1 implies F.n misses F.m proof assume 1 < n & m = 1; then F.n = {} & F.m = B by A5; then F.n /\ F.m = {}; hence thesis by XBOOLE_0:def 7; end; 1 < n & 1 < m implies F.n misses F.m proof assume 1 < n & 1 < m; then F.n = {} & F.m = {} by A5; then F.n /\ F.m = {}; hence thesis by XBOOLE_0:def 7; end; hence thesis by A2,A5,A7,A8,A9,A10,A11,A12; end; for m,n being set st m <> n holds F.m misses F.n proof let m,n be set; assume A13: m <> n; per cases; suppose m in NAT & n in NAT; hence thesis by A6,A13; suppose not m in NAT or not n in NAT; then not m in dom F or not n in dom F by FUNCT_2:def 1; then F.m = {} or F.n = {} by FUNCT_1:def 4; hence thesis by XBOOLE_1:65; end; then reconsider F as disjoint_valued Function of NAT, S by PROB_2:def 3; A14:union rng F = A \/ B proof union {A,B,{}} = union {A,B} by Th1; hence thesis by A5,ZFMISC_1:93; end; A15:for r being Nat holds (M*F).0 = M.A & (M*F).1 = M.B & (1+1 <= r implies (M*F).r = 0.) proof let r be Nat; A16:for r being Nat holds (M*F).r = M.(F.r) proof let r be Nat; dom (M*F) = NAT by FUNCT_2:def 1; hence thesis by FUNCT_1:22; end; 1 + 1 <= r implies (M*F).r = 0. proof assume 1 + 1 <= r; then 1 < r by NAT_1:38; then F.r = {} by A5; hence thesis by A1,A16; end; hence thesis by A5,A16; end; set H = M*F; A17:H is nonnegative by A1,Th54; A18: 0 + 1 = 0 + 1; set y1 = Ser H.1; set n2 = 1 + 1; reconsider A,B as Element of S; Ser H.n2 = y1 + H.n2 by SUPINF_2:63; then Ser H.n2 = y1 + 0.by A15 .= Ser H.1 by SUPINF_2:18 .= Ser H.0 + H.1 by A18,SUPINF_2:63 .= M.A + M.B by A15,SUPINF_2:63; then SUM(M*F) = M.A + M.B by A15,A17,SUPINF_2:67; hence thesis by A14,Def11; end; hence thesis by A1,Def5; end; theorem for S being sigma_Field_Subset of X, M being sigma_Measure of S, A,B being Element of S st A misses B holds M.(A \/ B) = M.A + M.B proof let S be sigma_Field_Subset of X, M be sigma_Measure of S, A,B be Element of S; assume A1:A misses B; M is Measure of S by Th60; hence thesis by A1,Def5; end; theorem Th62: for S being sigma_Field_Subset of X, M being sigma_Measure of S, A,B being Element of S st A c= B holds M.A <=' M.B proof let S be sigma_Field_Subset of X, M be sigma_Measure of S, A,B be Element of S; assume A1:A c= B; M is Measure of S by Th60; hence thesis by A1,Th25; end; theorem for S being sigma_Field_Subset of X, M being sigma_Measure of S, A,B being Element of S st A c= B & M.A <' +infty holds M.(B \ A) = M.B - M.A proof let S be sigma_Field_Subset of X, M be sigma_Measure of S, A,B be Element of S; assume A1:A c= B & M.A <' +infty; M is Measure of S by Th60; hence thesis by A1,Th26; end; theorem Th64: for S being sigma_Field_Subset of X, M being sigma_Measure of S, A,B being Element of S holds M.(A \/ B) <=' M.A + M.B proof let S be sigma_Field_Subset of X, M be sigma_Measure of S, A,B be Element of S; M is Measure of S by Th60; hence thesis by Th27; end; definition let X be set, S be sigma_Field_Subset of X, M be sigma_Measure of S, A be set; pred A is_measurable M means :Def12:A in S; end; canceled; theorem for S being sigma_Field_Subset of X, M being sigma_Measure of S holds {} is_measurable M & X is_measurable M & for A,B being set st A is_measurable M & B is_measurable M holds X \ A is_measurable M & A \/ B is_measurable M & A /\ B is_measurable M proof let S be sigma_Field_Subset of X, M be sigma_Measure of S; A1:{} in S & X in S by Th45; for A,B being set st A is_measurable M & B is_measurable M holds X \ A is_measurable M & A \/ B is_measurable M & A /\ B is_measurable M proof let A,B be set; assume A is_measurable M & B is_measurable M; then A in S & B in S by Def12; then X \ A in S & A \/ B in S & A /\ B in S & A \ B in S by Def3,Th46, Th47; hence thesis by Def12; end; hence thesis by A1,Def12; end; theorem for S being sigma_Field_Subset of X, M being sigma_Measure of S, T being N_Sub_set_fam of X st (for A being set st A in T holds A is_measurable M) holds union T is_measurable M & meet T is_measurable M proof let S be sigma_Field_Subset of X, M be sigma_Measure of S, T be N_Sub_set_fam of X; assume A1:for A being set st A in T holds A is_measurable M; A2:T c= S proof let x be set; assume x in T; then x is_measurable M by A1; hence thesis by Def12; end; then A3:union T in S by Def9; meet T in S by A2,Th49; hence thesis by A3,Def12; end; definition let X be set, S be sigma_Field_Subset of X, M be sigma_Measure of S; mode measure_zero of M -> Element of S means :Def13:M.it = 0.; existence proof {} is Element of S by Th45; then consider A being Element of S such that A1:A = {}; take A; thus thesis by A1,Def11; end; end; canceled; theorem for S being sigma_Field_Subset of X, M being sigma_Measure of S, A being Element of S, B being measure_zero of M st A c= B holds A is measure_zero of M proof let S be sigma_Field_Subset of X, M be sigma_Measure of S, A be Element of S, B be measure_zero of M; assume A1:A c= B; M is nonnegative by Def11; then 0.<=' M.A & M.A <=' M.B by A1,Def4,Th62; then 0.<=' M.A & M.A <=' 0.by Def13; then M.A = 0.by SUPINF_1:22; hence thesis by Def13; end; theorem for S being sigma_Field_Subset of X, M being sigma_Measure of S, A,B being measure_zero of M holds A \/ B is measure_zero of M & A /\ B is measure_zero of M & A \ B is measure_zero of M proof let S be sigma_Field_Subset of X, M be sigma_Measure of S, A,B be measure_zero of M; A1:M.A = 0.& M.B = 0.by Def13; M is nonnegative by Def11; then A2: 0.<=' M.(A \/ B) & 0.<=' M.(A /\ B) & 0.<=' M.(A \ B) by Def4; M.(A \/ B) <=' 0.+ 0.by A1,Th64; then M.(A \/ B) <=' 0.by SUPINF_2:18; then A3: M.(A \/ B) = 0.by A2,SUPINF_1:22; A /\ B c= A by XBOOLE_1:17; then M.(A /\ B) <=' 0.by A1,Th62; then A4: M.(A /\ B) = 0.by A2,SUPINF_1:22; A \ B c= A by XBOOLE_1:36; then M.(A \ B) <=' 0.by A1,Th62; then M.(A \ B) = 0.by A2,SUPINF_1:22; hence thesis by A3,A4,Def13; end; theorem for S being sigma_Field_Subset of X, M being sigma_Measure of S, A being Element of S, B being measure_zero of M holds M.(A \/ B) = M.A & M.(A /\ B) = 0.& M.(A \ B) = M.A proof let S be sigma_Field_Subset of X, M be sigma_Measure of S, A be Element of S, B be measure_zero of M; A1:M.B = 0.by Def13; M is nonnegative by Def11; then A2: 0.<=' M.(A \/ B) & 0.<=' M.(A /\ B) & 0.<=' M.(A \ B) by Def4; M.(A \/ B) <=' M.A + 0.by A1,Th64; then A3:M.(A \/ B) <=' M.A by SUPINF_2:18; A c= A \/ B by XBOOLE_1:7; then A4: M.A <=' M.(A \/ B) by Th62; A /\ B c= B by XBOOLE_1:17; then A5:M.(A /\ B) <=' 0.by A1,Th62; then A6:M.(A /\ B) = 0.by A2,SUPINF_1:22; A \ B c= A by XBOOLE_1:36; then A7:M.(A \ B) <=' M.A by Th62; M.A = M.((A /\ B) \/ (A \ B)) by XBOOLE_1:51; then M.A <=' 0.+ M.(A \ B) by A6,Th64; then M.A <=' M.(A \ B) by SUPINF_2:18; hence thesis by A2,A3,A4,A5,A7,SUPINF_1:22; end;

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