Volume 4, 1992

University of Bialystok

Copyright (c) 1992 Association of Mizar Users

### The abstract of the Mizar article:

### Properties of Caratheodor's Measure

**by****Jozef Bialas**- Received June 25, 1992
- MML identifier: MEASURE4

- [ Mizar article, MML identifier index ]

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; 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 :: MEASURE4:1 0.<=' x & 0.<=' y & 0.<=' z implies (x + y) + z = x + (y + z); theorem :: MEASURE4:2 (not x = -infty & not x = +infty) implies (y + x <=' z iff y <=' z - x); theorem :: MEASURE4:3 (0. <=' x & 0. <=' y) implies x + y = y + x; :: Some additional theorems about measures and functions theorem :: MEASURE4:4 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; theorem :: MEASURE4:5 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; theorem :: MEASURE4:6 bool X is sigma_Field_Subset of X; definition let X be set; let F be Function of NAT,bool X; redefine func rng F -> Subset-Family of X; end; definition let X be set; let A be Subset-Family of X; redefine func union A -> Element of bool X; 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; end; theorem :: MEASURE4:7 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)); theorem :: MEASURE4:8 ex M being Function of bool X,ExtREAL st for A being Element of bool X holds M.A = 0.; canceled 2; theorem :: MEASURE4:11 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); definition let X be set; canceled; mode C_Measure of X -> Function of bool X,ExtREAL means :: MEASURE4:def 2 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); 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 :: MEASURE4:def 3 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))); end; theorem :: MEASURE4:12 for W,Z being Element of bool X holds C.(W \/ Z) <=' C.W + C.Z; theorem :: MEASURE4:13 for W,Z being Element of bool X holds C.Z + C.W = C.W + C.Z; theorem :: MEASURE4:14 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))); theorem :: MEASURE4:15 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; theorem :: MEASURE4:16 A in sigma_Field(C) implies X \ A in sigma_Field(C); theorem :: MEASURE4:17 A in sigma_Field(C) & B in sigma_Field(C) implies A \/ B in sigma_Field(C); theorem :: MEASURE4:18 A in sigma_Field(C) & B in sigma_Field(C) implies A /\ B in sigma_Field(C); theorem :: MEASURE4:19 A in sigma_Field(C) & B in sigma_Field(C) implies A \ B in sigma_Field(C); theorem :: MEASURE4:20 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; theorem :: MEASURE4:21 sigma_Field(C) is sigma_Field_Subset of X; definition let X be set; let C be C_Measure of X; cluster sigma_Field(C) -> sigma_Field_Subset-like compl-closed non empty; 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; end; definition let X be set; let C be C_Measure of X; func sigma_Meas(C) -> Function of sigma_Field(C),ExtREAL means :: MEASURE4:def 4 for A being Element of bool X st A in sigma_Field(C) holds it.A = C.A; end; theorem :: MEASURE4:22 sigma_Meas(C) is Measure of sigma_Field(C); 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; end; theorem :: MEASURE4:23 sigma_Meas(C) is sigma_Measure of sigma_Field(C); definition let X be set; let C be C_Measure of X; redefine func sigma_Meas(C) -> sigma_Measure of sigma_Field(C); end; theorem :: MEASURE4:24 for A being Element of bool X holds C.A = 0. implies A in sigma_Field(C); theorem :: MEASURE4:25 sigma_Meas(C) is_complete sigma_Field(C);

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