Volume 3, 1991

University of Bialystok

Copyright (c) 1991 Association of Mizar Users

### The abstract of the Mizar article:

### Several Properties of the $\sigma$-additive Measure

**by****Jozef Bialas**- Received July 3, 1991
- MML identifier: MEASURE2

- [ Mizar article, MML identifier index ]

environ vocabulary MEASURE1, FUNCT_1, SUPINF_2, BOOLE, SETFAM_1, TARSKI, RELAT_1, ARYTM_3, RLVECT_1, PROB_1, ORDINAL2, MEASURE2; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, NUMBERS, XREAL_0, RELAT_1, FUNCT_1, FUNCT_2, NAT_1, SETFAM_1, SUPINF_1, SUPINF_2, MEASURE1; constructors NAT_1, SUPINF_2, MEASURE1, PROB_2, MEMBERED, XBOOLE_0; clusters SUBSET_1, SUPINF_1, MEASURE1, RELSET_1, ARYTM_3, MEMBERED, ZFMISC_1, XBOOLE_0, NUMBERS, ORDINAL2; requirements NUMERALS, SUBSET, BOOLE; begin :: :: Some useful theorems about measures and functions :: reserve X for set; theorem :: MEASURE2:1 for S being sigma_Field_Subset of X, M being sigma_Measure of S, F being Function of NAT,S holds M*F is nonnegative; definition let X be set; let S be sigma_Field_Subset of X; mode N_Measure_fam of S -> N_Sub_set_fam of X means :: MEASURE2:def 1 it c= S; end; canceled; theorem :: MEASURE2:3 for S being sigma_Field_Subset of X, T being N_Measure_fam of S holds meet T in S & union T in S; definition let X be set; let S be sigma_Field_Subset of X; let T be N_Measure_fam of S; redefine func meet T -> Element of S; redefine func union T -> Element of S; end; theorem :: MEASURE2:4 for S being sigma_Field_Subset of X, N being Function of NAT,S holds ex F being Function of NAT,S st F.0 = N.0 & for n being Element of NAT holds F.(n+1) = N.(n+1) \ N.n; theorem :: MEASURE2:5 for S being sigma_Field_Subset of X, N being Function of NAT,S holds ex F being Function of NAT,S st F.0 = N.0 & for n being Element of NAT holds F.(n+1) = N.(n+1) \/ F.n; theorem :: MEASURE2:6 for S being non empty Subset-Family of X, N,F being Function of NAT,S holds F.0 = N.0 & (for n being Element of NAT holds F.(n+1) = N.(n+1) \/ F.n) implies (for r being set for n being Nat holds (r in F.n iff ex k being Nat st (k <= n & r in N.k))); theorem :: MEASURE2:7 for S being non empty Subset-Family of X, N,F being Function of NAT,S holds (F.0 = N.0 & (for n being Element of NAT holds F.(n+1) = N.(n+1) \/ F.n) implies for n,m being Nat st n < m holds F.n c= F.m); theorem :: MEASURE2:8 for S being non empty Subset-Family of X, N, G, F being Function of NAT,S holds (G.0 = N.0 & (for n being Element of NAT holds G.(n+1) = N.(n+1) \/ G.n) & F.0 = N.0 & (for n being Element of NAT holds F.(n+1) = N.(n+1) \ G.n)) implies (for n,m being Nat st n <= m holds F.n c= G.m); theorem :: MEASURE2:9 for S being sigma_Field_Subset of X holds for N, G being Function of NAT,S holds ex F being Function of NAT,S st F.0 = N.0 & for n being Element of NAT holds F.(n+1) = N.(n+1) \ G.n; theorem :: MEASURE2:10 for S being sigma_Field_Subset of X holds for N being Function of NAT,S holds ex F being Function of NAT,S st F.0 = {} & for n being Element of NAT holds F.(n+1) = N.0 \ N.n; theorem :: MEASURE2:11 for S being sigma_Field_Subset of X holds for N, G, F being Function of NAT,S holds (G.0 = N.0 & (for n being Element of NAT holds G.(n+1) = N.(n+1) \/ G.n) & F.0 = N.0 & (for n being Element of NAT holds F.(n+1) = N.(n+1) \ G.n)) implies (for n,m being Nat st n < m holds F.n misses F.m); canceled; theorem :: MEASURE2:13 for S being sigma_Field_Subset of X, M being sigma_Measure of S, T being N_Measure_fam of S, F being Function of NAT,S st T = rng F holds M.(union T) <=' SUM(M*F); theorem :: MEASURE2:14 for S being sigma_Field_Subset of X, T being N_Measure_fam of S holds ex F being Function of NAT,S st T = rng F; theorem :: MEASURE2:15 for S being sigma_Field_Subset of X, N, F being Function of NAT,S st (F.0 = {} & for n being Element of NAT holds (F.(n+1) = N.0 \ N.n & N.(n+1) c= N.n)) holds for n being Element of NAT holds F.n c= F.(n+1); theorem :: MEASURE2:16 for S being sigma_Field_Subset of X, M being sigma_Measure of S, T being N_Measure_fam of S holds (for A being set holds A in T implies A is measure_zero of M) implies union T is measure_zero of M; theorem :: MEASURE2:17 for S being sigma_Field_Subset of X, M being sigma_Measure of S, T being N_Measure_fam of S st (ex A being set st A in T & A is measure_zero of M) holds meet T is measure_zero of M; theorem :: MEASURE2:18 for S being sigma_Field_Subset of X, M being sigma_Measure of S, T being N_Measure_fam of S st (for A being set holds A in T implies A is measure_zero of M) holds meet T is measure_zero of M; definition let X be set; let S be sigma_Field_Subset of X; let IT be N_Measure_fam of S; attr IT is non-decreasing means :: MEASURE2:def 2 ex F being Function of NAT,S st IT = rng F & for n being Element of NAT holds F.n c= F.(n+1); end; definition let X be set; let S be sigma_Field_Subset of X; cluster non-decreasing N_Measure_fam of S; end; definition let X be set; let S be sigma_Field_Subset of X; let IT be N_Measure_fam of S; attr IT is non-increasing means :: MEASURE2:def 3 ex F being Function of NAT,S st (IT = rng F & for n being Element of NAT holds F.(n+1) c= F.n); end; definition let X be set; let S be sigma_Field_Subset of X; cluster non-increasing N_Measure_fam of S; end; canceled 2; theorem :: MEASURE2:21 for S being sigma_Field_Subset of X, N,F being Function of NAT,S holds (F.0 = {} & for n being Element of NAT holds (F.(n+1) = N.0 \ N.n & N.(n+1) c= N.n)) implies rng F is non-decreasing N_Measure_fam of S; theorem :: MEASURE2:22 for S being non empty Subset-Family of X, N being Function of NAT,S holds (for n being Element of NAT holds N.n c= N.(n+1)) implies (for m,n being Nat st n < m holds N.n c= N.m); theorem :: MEASURE2:23 for S being sigma_Field_Subset of X, N,F being Function of NAT,S holds (F.0 = N.0 & for n being Element of NAT holds (F.(n+1) = N.(n+1) \ N.n & N.n c= N.(n+1))) implies (for n,m being Nat st n < m holds F.n misses F.m); theorem :: MEASURE2:24 for S being sigma_Field_Subset of X, N,F being Function of NAT,S holds (F.0 = N.0 & for n being Element of NAT holds (F.(n+1) = N.(n+1) \ N.n & N.n c= N.(n+1))) implies union rng F = union rng N; theorem :: MEASURE2:25 for S being sigma_Field_Subset of X, N,F being Function of NAT,S holds (F.0 = N.0 & for n being Element of NAT holds (F.(n+1) = N.(n+1) \ N.n & N.n c= N.(n+1))) implies F is Sep_Sequence of S; theorem :: MEASURE2:26 for S being sigma_Field_Subset of X, N,F being Function of NAT,S holds (F.0 = N.0 & for n being Element of NAT holds (F.(n+1) = N.(n+1) \ N.n & N.n c= N.(n+1))) implies (N.0 = F.0 & for n being Element of NAT holds N.(n+1) = F.(n+1) \/ N.n); theorem :: MEASURE2:27 for S being sigma_Field_Subset of X, M being sigma_Measure of S, F being Function of NAT,S st (for n being Element of NAT holds F.n c= F.(n+1)) holds M.(union rng F) = sup(rng (M*F));

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