Volume 7, 1995

University of Bialystok

Copyright (c) 1995 Association of Mizar Users

### The abstract of the Mizar article:

### The One-Dimensional Lebesgue Measure

**by****Jozef Bialas**- Received February 4, 1995
- MML identifier: MEASURE7

- [ Mizar article, MML identifier index ]

environ vocabulary FUNCT_1, SUPINF_1, RLVECT_1, SUPINF_2, ORDINAL2, RELAT_1, ARYTM_3, TARSKI, MEASURE5, FUNCT_2, BOOLE, COMPLEX1, FUNCT_3, MCART_1, MEASURE4, MEASURE1, MEASURE3, MEASURE7; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, MCART_1, NUMBERS, XREAL_0, REAL_1, RELAT_1, FUNCT_1, NAT_1, SUPINF_1, SUPINF_2, MEASURE1, MEASURE3, MEASURE4, MEASURE5, FUNCT_2, DTCONSTR; constructors NAT_1, SUPINF_2, MEASURE3, MEASURE4, MEASURE6, DTCONSTR, MCART_1, FRAENKEL, MEMBERED, XBOOLE_0; clusters SUBSET_1, SUPINF_1, MEASURE4, RELSET_1, ARYTM_3, FRAENKEL, FUNCT_2, MEMBERED, ZFMISC_1, XBOOLE_0, ORDINAL2; requirements NUMERALS, BOOLE, SUBSET, ARITHM; begin :: Some theorems about series of R_eal numbers theorem :: MEASURE7:1 for F being Function of NAT,ExtREAL st for n being Nat holds F.n = 0. holds SUM(F) = 0.; theorem :: MEASURE7:2 for F being Function of NAT,ExtREAL st F is nonnegative holds for n being Nat holds F.n <=' Ser(F).n; theorem :: MEASURE7:3 for F,G,H being Function of NAT,ExtREAL st G is nonnegative & H is nonnegative holds (for n being Nat holds F.n = G.n + H.n ) implies (for n being Nat holds (Ser F).n = (Ser G).n + (Ser H).n ); theorem :: MEASURE7:4 for F,G,H being Function of NAT,ExtREAL st for n being Nat holds F.n = G.n + H.n holds G is nonnegative & H is nonnegative implies SUM(F) <=' SUM(G) + SUM(H); canceled; theorem :: MEASURE7:6 for F,G being Function of NAT,ExtREAL holds (F is nonnegative & for n being Nat holds F.n <=' G.n) implies for n being Nat holds (Ser(F)).n <=' SUM(G); theorem :: MEASURE7:7 for F being Function of NAT,ExtREAL holds F is nonnegative implies for n being Nat holds (Ser(F)).n <=' SUM(F); definition let S be non empty Subset of NAT; let H be Function of S,NAT; let n be Element of S; redefine func H.n -> Nat; end; definition let S be non empty set; let H be Function of S,ExtREAL; func On H -> Function of NAT,ExtREAL means :: MEASURE7:def 1 for n being Element of NAT holds (n in S implies it.n = H.n) & (not n in S implies it.n = 0.); end; theorem :: MEASURE7:8 for X being non empty set for G being Function of X,ExtREAL st G is nonnegative holds On G is nonnegative; theorem :: MEASURE7:9 for F being Function of NAT,ExtREAL st F is nonnegative holds for n,k being Nat st n <= k holds Ser(F).n <=' Ser(F).k; theorem :: MEASURE7:10 for k being Nat holds for F being Function of NAT,ExtREAL holds ((for n being Nat st n <> k holds F.n = 0.) implies ((for n being Nat st n < k holds Ser(F).n = 0.) & (for n being Nat st k <= n holds Ser(F).n = F.k))); theorem :: MEASURE7:11 for G being Function of NAT,ExtREAL st G is nonnegative holds for S being non empty Subset of NAT holds for H being Function of S,NAT st H is one-to-one holds SUM(On(G*H)) <=' SUM(G); theorem :: MEASURE7:12 for F,G being Function of NAT,ExtREAL st F is nonnegative & G is nonnegative holds for S being non empty Subset of NAT holds for H being Function of S,NAT st H is one-to-one holds (for k being Nat holds ((k in S implies F.k = (G*H).k) & ((not k in S) implies F.k = 0.))) implies SUM(F) <=' SUM(G); definition let A be Subset of REAL; mode Interval_Covering of A -> Function of NAT, bool REAL means :: MEASURE7:def 2 A c= union(rng it) & for n being Nat holds it.n is Interval; end; definition let A be Subset of REAL; let F be Interval_Covering of A; let n be Nat; redefine func F.n ->Interval; end; definition let F be Function of NAT,bool REAL; mode Interval_Covering of F -> Function of NAT,Funcs(NAT,bool REAL) means :: MEASURE7:def 3 for n being Nat holds it.n is Interval_Covering of F.n; end; definition let A be Subset of REAL; let F be Interval_Covering of A; func F vol -> Function of NAT,ExtREAL means :: MEASURE7:def 4 for n being Nat holds it.n = vol(F.n); end; theorem :: MEASURE7:13 for A being Subset of REAL holds for F being Interval_Covering of A holds (F vol) is nonnegative; definition let F be Function of NAT, bool REAL; let H be Interval_Covering of F; let n be Nat; redefine func H.n -> Interval_Covering of F.n; end; definition let F be Function of NAT, bool REAL; let G be Interval_Covering of F; func G vol -> Function of NAT,Funcs(NAT,ExtREAL) means :: MEASURE7:def 5 for n being Nat holds it.n = (G.n) vol; end; definition let A be Subset of REAL; let F be Interval_Covering of A; func vol(F) -> R_eal equals :: MEASURE7:def 6 SUM(F vol); end; definition let F be Function of NAT,(bool REAL); let G be Interval_Covering of F; func vol(G) -> Function of NAT,ExtREAL means :: MEASURE7:def 7 for n being Nat holds it.n = vol(G.n); end; theorem :: MEASURE7:14 for F being Function of NAT,(bool REAL) holds for G being Interval_Covering of F holds for n being Nat holds 0. <=' (vol(G)).n; definition let A be Subset of REAL; func Svc(A) -> Subset of ExtREAL means :: MEASURE7:def 8 for x being R_eal holds x in it iff ex F being Interval_Covering of A st x = vol(F); end; definition let A be Subset of REAL; cluster Svc(A) -> non empty; end; definition let A be Element of bool REAL; func COMPLEX(A) -> Element of ExtREAL equals :: MEASURE7:def 9 inf(Svc(A)); end; definition func OS_Meas -> Function of bool REAL,ExtREAL means :: MEASURE7:def 10 for A being Subset of REAL holds it.A = inf(Svc(A)); end; definition let H be Function of NAT,[:NAT,NAT:]; redefine func pr1(H) -> Function of NAT,NAT means :: MEASURE7:def 11 for n being Element of NAT holds ex s being Element of NAT st H.n = [it.n,s]; end; definition let H be Function of NAT,[:NAT,NAT:]; redefine func pr2(H) -> Function of NAT,NAT means :: MEASURE7:def 12 for n being Element of NAT holds H.n = [pr1(H).n,it.n]; end; definition let F be Function of NAT,bool REAL; let G be Interval_Covering of F; let H be Function of NAT,[:NAT,NAT:] such that rng H = [:NAT,NAT:]; func On(G,H) -> Interval_Covering of union rng F means :: MEASURE7:def 13 for n being Element of NAT holds it.n = (G.(pr1(H).n)).(pr2(H).n); end; theorem :: MEASURE7:15 for H being Function of NAT,[:NAT,NAT:] st H is one-to-one & rng H = [:NAT,NAT:] holds for k being Nat holds ex m being Nat st for F being Function of NAT,bool REAL holds for G being Interval_Covering of F holds Ser((On(G,H)) vol).k <=' Ser(vol(G)).m; theorem :: MEASURE7:16 for F being Function of NAT,bool REAL holds for G being Interval_Covering of F holds inf Svc(union rng F) <=' SUM(vol(G)); theorem :: MEASURE7:17 OS_Meas is C_Measure of REAL; definition redefine func OS_Meas -> C_Measure of REAL; end; definition func Lmi_sigmaFIELD -> sigma_Field_Subset of REAL equals :: MEASURE7:def 14 sigma_Field(OS_Meas); end; definition func L_mi -> sigma_Measure of Lmi_sigmaFIELD equals :: MEASURE7:def 15 sigma_Meas(OS_Meas); end; theorem :: MEASURE7:18 L_mi is_complete Lmi_sigmaFIELD; canceled 2; theorem :: MEASURE7:21 for A being set st A in Lmi_sigmaFIELD holds REAL \ A in Lmi_sigmaFIELD;

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