Volume 12, 2000

University of Bialystok

Copyright (c) 2000 Association of Mizar Users

### The abstract of the Mizar article:

### Definitions and Basic Properties of Measurable Functions

**by****Noboru Endou,****Katsumi Wasaki, and****Yasunari Shidama**- Received September 7, 2000
- MML identifier: MESFUNC1

- [ Mizar article, MML identifier index ]

environ vocabulary INT_1, RAT_1, ARYTM_1, TARSKI, FUNCT_1, FUNCT_2, RELAT_1, BOOLE, ARYTM_3, CARD_4, CARD_1, FINSET_1, PARTFUN1, SUPINF_1, SEQ_1, MEASURE6, RLVECT_1, GROUP_1, COMPLEX1, MEASURE1, SETFAM_1, MESFUNC1, ARYTM; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0, XREAL_0, MEASURE6, REAL_1, RELAT_1, FUNCT_1, NAT_1, INT_1, RAT_1, FINSET_1, CARD_1, CARD_4, SUPINF_1, SUPINF_2, MEASURE1, MEASURE2, MEASURE3, FUNCT_2, PARTFUN1, EXTREAL1; constructors NAT_1, MEASURE3, MEASURE6, RAT_1, WELLORD2, EXTREAL1, REAL_1, SUPINF_2, MEMBERED; clusters SUBSET_1, SUPINF_1, MEASURE1, XREAL_0, RELSET_1, INT_1, CARD_1, NAT_1, MEMBERED, NUMBERS, ORDINAL2; requirements NUMERALS, BOOLE, SUBSET, REAL, ARITHM; begin :: Cardinal numbers of INT and RAT reserve k for Nat; reserve r for Real; reserve i for Integer; reserve q for Rational; definition func INT- -> Subset of REAL means :: MESFUNC1:def 1 r in it iff ex k st r = - k; end; definition cluster INT- -> non empty; end; theorem :: MESFUNC1:1 NAT,INT- are_equipotent; theorem :: MESFUNC1:2 INT=INT- \/ NAT; theorem :: MESFUNC1:3 NAT,INT are_equipotent; definition redefine func INT -> Subset of REAL; end; definition let n be Nat; func RAT_with_denominator n -> Subset of RAT means :: MESFUNC1:def 2 q in it iff ex i st q = i/n; end; definition let n be Nat; cluster RAT_with_denominator(n+1) -> non empty; end; theorem :: MESFUNC1:4 for n being Nat holds INT,RAT_with_denominator (n+1) are_equipotent; theorem :: MESFUNC1:5 NAT,RAT are_equipotent; begin :: Basic operations on R_EAL valued functions definition let C be non empty set, f be PartFunc of C, ExtREAL, x be set; redefine func f.x -> R_eal; end; definition let C be non empty set; let f1,f2 be PartFunc of C,ExtREAL; func f1+f2 -> PartFunc of C,ExtREAL means :: MESFUNC1:def 3 dom it = (dom f1 /\ dom f2)\((f1"{-infty} /\ f2"{+infty}) \/ (f1"{+infty} /\ f2"{-infty})) & for c being Element of C st c in dom it holds it.c = f1.c + f2.c; func f1-f2 -> PartFunc of C,ExtREAL means :: MESFUNC1:def 4 dom it = (dom f1 /\ dom f2)\((f1"{+infty} /\ f2"{+infty}) \/ (f1"{-infty} /\ f2"{-infty})) & for c being Element of C st c in dom it holds it.c = f1.c - f2.c; func f1(#)f2 -> PartFunc of C,ExtREAL means :: MESFUNC1:def 5 dom it = dom f1 /\ dom f2 & for c being Element of C st c in dom it holds it.c = f1.c * f2.c; end; definition let C be non empty set, f be PartFunc of C,ExtREAL, r be Real; func r(#)f -> PartFunc of C,ExtREAL means :: MESFUNC1:def 6 dom it = dom f & for c being Element of C st c in dom it holds it.c = (R_EAL r) * f.c; end; theorem :: MESFUNC1:6 for C being non empty set, f being PartFunc of C,ExtREAL, r being Real st r <> 0 holds for c being Element of C st c in dom(r(#)f) holds f.c = (r(#)f).c / R_EAL r; definition let C be non empty set; let f be PartFunc of C,ExtREAL; func -f -> PartFunc of C,ExtREAL means :: MESFUNC1:def 7 dom it = dom f & for c being Element of C st c in dom it holds it.c = -(f.c); end; definition func 1. -> R_eal equals :: MESFUNC1:def 8 1; end; definition let C be non empty set; let f be PartFunc of C,ExtREAL; let r be Real; func r/f -> PartFunc of C,ExtREAL means :: MESFUNC1:def 9 dom it = dom f \ f"{0.} & for c being Element of C st c in dom it holds it.c = (R_EAL r)/(f.c); end; theorem :: MESFUNC1:7 for C being non empty set, f being PartFunc of C,ExtREAL holds dom (1/f) = dom f \ f"{0.} & for c being Element of C st c in dom (1/f) holds (1/f).c = 1./(f.c); definition let C be non empty set; let f be PartFunc of C,ExtREAL; func |.f.| -> PartFunc of C,ExtREAL means :: MESFUNC1:def 10 dom it = dom f & for c being Element of C st c in dom it holds it.c = |. f.c .|; end; canceled; theorem :: MESFUNC1:9 for C being non empty set, f1,f2 being PartFunc of C,ExtREAL holds f1 + f2 = f2 + f1; theorem :: MESFUNC1:10 for C being non empty set, f1,f2 being PartFunc of C,ExtREAL holds f1 (#) f2 = f2 (#) f1; definition let C be non empty set; let f1,f2 be PartFunc of C,ExtREAL; redefine func f1+f2; commutativity; redefine func f1(#)f2; commutativity; end; begin :: Level sets theorem :: MESFUNC1:11 for r being Real holds ex n being Nat st r <= n; theorem :: MESFUNC1:12 for r being Real holds ex n being Nat st -n <= r; theorem :: MESFUNC1:13 for r,s being real number st r < s holds ex n being Nat st 1/(n+1) < s-r; theorem :: MESFUNC1:14 for r,s being real number st for n being Nat holds r-1/(n+1) <= s holds r <= s; theorem :: MESFUNC1:15 for a being R_eal st (for r being Real holds R_EAL r <' a) holds a = +infty; theorem :: MESFUNC1:16 for a being R_eal st (for r being Real holds a <' R_EAL r) holds a = -infty; definition let X be set; let S be sigma_Field_Subset of X; let A be set; pred A is_measurable_on S means :: MESFUNC1:def 11 A in S; end; theorem :: MESFUNC1:17 for X,A being set, S being sigma_Field_Subset of X holds A is_measurable_on S iff (for M being sigma_Measure of S holds A is_measurable M); reserve X for non empty set; reserve x for Element of X; reserve f,g for PartFunc of X,ExtREAL; reserve S for sigma_Field_Subset of X; reserve F for Function of NAT,S; reserve A for set; reserve a for R_eal; reserve r,s for Real; reserve n,m for Nat; definition let X,f,a; func less_dom(f,a) -> Subset of X means :: MESFUNC1:def 12 x in it iff x in dom f & ex y being R_eal st y=f.x & y <' a; func less_eq_dom(f,a) -> Subset of X means :: MESFUNC1:def 13 x in it iff x in dom f & ex y being R_eal st y=f.x & y <=' a; func great_dom(f,a) -> Subset of X means :: MESFUNC1:def 14 x in it iff x in dom f & ex y being R_eal st y=f.x & a <' y; func great_eq_dom(f,a) -> Subset of X means :: MESFUNC1:def 15 x in it iff x in dom f & ex y being R_eal st y=f.x & a <=' y; func eq_dom(f,a) -> Subset of X means :: MESFUNC1:def 16 x in it iff x in dom f & ex y being R_eal st y=f.x & a=y; end; theorem :: MESFUNC1:18 for X, S, f, A, a st A c= dom f holds A /\ great_eq_dom(f,a) = A\(A /\ less_dom(f,a)); theorem :: MESFUNC1:19 for X, S, f, A, a st A c= dom f holds A /\ great_dom(f,a) = A\(A /\ less_eq_dom(f,a)); theorem :: MESFUNC1:20 for X, S, f, A, a st A c= dom f holds A /\ less_eq_dom(f,a) = A\(A /\ great_dom(f,a)); theorem :: MESFUNC1:21 for X, S, f, A, a st A c= dom f holds A /\ less_dom(f,a) = A\(A /\ great_eq_dom(f,a)); theorem :: MESFUNC1:22 for X, S, f, A, a holds A /\ eq_dom(f,a) = A /\ great_eq_dom(f,a) /\ less_eq_dom(f,a); theorem :: MESFUNC1:23 for X, S, F, f, A, r st for n holds F.n = A /\ great_dom(f,R_EAL(r-1/(n+1))) holds A /\ great_eq_dom(f,R_EAL r) = meet rng F; theorem :: MESFUNC1:24 for X, S, F, f, A for r being real number st for n holds F.n = A /\ less_dom(f,R_EAL(r+1/(n+1))) holds A /\ less_eq_dom(f,R_EAL r) = meet rng F; theorem :: MESFUNC1:25 for X, S, F, f, A for r being real number st for n holds F.n = A /\ less_eq_dom(f,R_EAL(r-1/(n+1))) holds A /\ less_dom(f,R_EAL r) = union rng F; theorem :: MESFUNC1:26 for X, S, F, f, A, r st for n holds F.n = A /\ great_eq_dom(f,R_EAL(r+1/(n+1))) holds A /\ great_dom(f,R_EAL r) = union rng F; theorem :: MESFUNC1:27 for X, S, F, f, A st for n holds F.n = A /\ great_dom(f,R_EAL n) holds A /\ eq_dom(f,+infty) = meet rng F; theorem :: MESFUNC1:28 for X, S, F, f, A st for n holds F.n = A /\ less_dom(f,R_EAL n) holds A /\ less_dom(f,+infty) = union rng F; theorem :: MESFUNC1:29 for X, S, F, f, A st for n holds F.n = A /\ less_dom(f,R_EAL (-n)) holds A /\ eq_dom(f,-infty) = meet rng F; theorem :: MESFUNC1:30 for X, S, F, f, A st for n holds F.n = A /\ great_dom(f,R_EAL (-n)) holds A /\ great_dom(f,-infty) = union rng F; begin :: Measurable functions definition let X be non empty set; let S be sigma_Field_Subset of X; let f be PartFunc of X,ExtREAL; let A be Element of S; pred f is_measurable_on A means :: MESFUNC1:def 17 for r being real number holds A /\ less_dom(f,R_EAL r) is_measurable_on S; end; reserve A,B for Element of S; theorem :: MESFUNC1:31 for X,S,f,A st A c= dom f holds f is_measurable_on A iff (for r being real number holds A /\ great_eq_dom(f,R_EAL r) is_measurable_on S) ; theorem :: MESFUNC1:32 for X,S,f,A holds f is_measurable_on A iff (for r being real number holds A /\ less_eq_dom(f,R_EAL r) is_measurable_on S); theorem :: MESFUNC1:33 for X,S,f,A st A c= dom f holds f is_measurable_on A iff (for r being real number holds A /\ great_dom(f,R_EAL r) is_measurable_on S); theorem :: MESFUNC1:34 for X,S,f,A,B st B c= A & f is_measurable_on A holds f is_measurable_on B; theorem :: MESFUNC1:35 for X,S,f,A,B st f is_measurable_on A & f is_measurable_on B holds f is_measurable_on (A \/ B); theorem :: MESFUNC1:36 for X,S,f,A,r,s st f is_measurable_on A & A c= dom f holds (A /\ great_dom(f,R_EAL r) /\ less_dom(f,R_EAL s)) is_measurable_on S; theorem :: MESFUNC1:37 for X,S,f,A st f is_measurable_on A & A c= dom f holds A /\ eq_dom(f,+infty) is_measurable_on S; theorem :: MESFUNC1:38 for X,S,f,A st f is_measurable_on A holds A /\ eq_dom(f,-infty) is_measurable_on S; theorem :: MESFUNC1:39 for X,S,f,A st f is_measurable_on A & A c= dom f holds A /\ great_dom(f,-infty) /\ less_dom(f,+infty) is_measurable_on S; theorem :: MESFUNC1:40 for X,S,f,g,A,r st f is_measurable_on A & g is_measurable_on A & A c= dom g holds A /\ less_dom(f,R_EAL r) /\ great_dom(g,R_EAL r) is_measurable_on S; theorem :: MESFUNC1:41 for X,S,f,A,r st f is_measurable_on A & A c= dom f holds r(#)f is_measurable_on A;

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