Volume 2, 1990

University of Bialystok

Copyright (c) 1990 Association of Mizar Users

### The abstract of the Mizar article:

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

**by****Jozef Bialas**- Received October 15, 1990
- MML identifier: MEASURE1

- [ Mizar article, 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; begin :: Some useful theorems about sets and R_eal numbers reserve X for set; theorem :: MEASURE1:1 for X,Y being set holds union {X,Y,{}} = union {X,Y}; canceled 2; theorem :: MEASURE1:4 for x,y,s,t being R_eal holds 0.<=' x & 0.<=' s & x <=' y & s <=' t implies x + s <=' y + t; theorem :: MEASURE1:5 for x,y,z being R_eal holds 0.<=' y & 0.<=' z & x = y + z & y <' +infty implies z = x - y; theorem :: MEASURE1:6 for A being Subset of X holds {A} is Subset-Family of X; theorem :: MEASURE1:7 for A,B being Subset of X holds {A,B} is Subset-Family of X; theorem :: MEASURE1:8 for A,B,C being Subset of X holds {A,B,C} is non empty Subset-Family of X; theorem :: MEASURE1:9 {{}} is Subset-Family of X; 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 ex B being set st B c= A() & P[B]; definition let X be set; let S be non empty Subset-Family of X; canceled; func X\S -> Subset-Family of X means :: MEASURE1:def 2 for A being set holds A in it iff ex B being set st B in S & A = X \ B; end; definition let X be set; let S be non empty Subset-Family of X; cluster X\S -> non empty; end; canceled 4; theorem :: MEASURE1:14 for S being non empty Subset-Family of X holds S = X \ (X \ S); theorem :: MEASURE1:15 for S being non empty Subset-Family of X holds meet S = X \ union (X \ S) & union S = X \ meet (X \ S); :: :: 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 :: MEASURE1:def 3 for A being set holds A in IT implies X\A in IT; end; theorem :: MEASURE1:16 for X being set, F being Subset-Family of X st F is cup-closed compl-closed holds F is cap-closed; definition let X be set; cluster cup-closed compl-closed -> cap-closed Subset-Family of X; cluster cap-closed compl-closed -> cup-closed Subset-Family of X; end; theorem :: MEASURE1:17 for S being Field_Subset of X holds S = X \ S; theorem :: MEASURE1:18 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; theorem :: MEASURE1:19 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; theorem :: MEASURE1:20 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; theorem :: MEASURE1:21 for S being Field_Subset of X holds {} in S & X in S; 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; end; definition let X be non empty set, F be Function of X,ExtREAL; redefine attr F is nonnegative means :: MEASURE1:def 4 for A being Element of X holds 0. <=' F.A; end; canceled; theorem :: MEASURE1:23 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; definition let X be set, S be Field_Subset of X; mode Measure of S -> Function of S,ExtREAL means :: MEASURE1:def 5 it is nonnegative & it.{} = 0. & for A,B being Element of S st A misses B holds it.(A \/ B) = it.A + it.B; end; canceled; theorem :: MEASURE1:25 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; theorem :: MEASURE1:26 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; definition let X be set; cluster non empty compl-closed cap-closed Subset-Family of X; 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; 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; redefine func A \ B -> Element of S; end; theorem :: MEASURE1:27 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; 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 :: MEASURE1:def 6 A in S; end; canceled; theorem :: MEASURE1:29 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; 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 :: MEASURE1:def 7 M.it = 0.; end; canceled; theorem :: MEASURE1:31 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; theorem :: MEASURE1:32 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; theorem :: MEASURE1:33 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; :: :: sigma_Field Subset of X and sigma_additive nonnegative measure :: theorem :: MEASURE1:34 for A being Subset of X ex F being Function of NAT,bool X st rng F = {A}; theorem :: MEASURE1:35 for A being set ex F being Function of NAT,{A} st for n being Nat holds F.n = A; definition let X be set; cluster non empty countable Subset-Family of X; 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 :: MEASURE1:38 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; theorem :: MEASURE1:39 for A,B being Subset of X holds {A,B,{}} is N_Sub_set_fam of X; theorem :: MEASURE1:40 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; theorem :: MEASURE1:41 for A,B being Subset of X holds {A,B} is N_Sub_set_fam of X; theorem :: MEASURE1:42 for S being N_Sub_set_fam of X holds X \ S is N_Sub_set_fam of X; definition let X be set; let IT be Subset-Family of X; canceled; attr IT is sigma_Field_Subset-like means :: MEASURE1:def 9 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; 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; canceled 2; theorem :: MEASURE1:45 for S being sigma_Field_Subset of X holds {} in S & X in S; definition let X be set; cluster -> non empty sigma_Field_Subset of X; end; theorem :: MEASURE1:46 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; theorem :: MEASURE1:47 for S being sigma_Field_Subset of X, A,B being set st A in S & B in S holds A \ B in S; theorem :: MEASURE1:48 for S being sigma_Field_Subset of X holds S = X \ S; theorem :: MEASURE1:49 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; definition let X be set; let S be sigma_Field_Subset of X; cluster disjoint_valued Function of NAT, S; 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; end; canceled 2; theorem :: MEASURE1:52 for S being sigma_Field_Subset of X, F being Function of NAT,S holds rng F is N_Sub_set_fam of X; theorem :: MEASURE1:53 for S being sigma_Field_Subset of X, F being Function of NAT,S holds union rng F is Element of S; theorem :: MEASURE1:54 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; theorem :: MEASURE1:55 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); theorem :: MEASURE1:56 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); theorem :: MEASURE1:57 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.; theorem :: MEASURE1:58 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); definition let X be set; let S be sigma_Field_Subset of X; canceled; mode sigma_Measure of S -> Function of S,ExtREAL means :: MEASURE1:def 11 it is nonnegative & it.{} = 0.& for F being Sep_Sequence of S holds SUM(it*F) = it.(union rng F); end; definition let X be set; cluster sigma_Field_Subset-like compl-closed -> cup-closed (non empty Subset-Family of X); end; canceled; theorem :: MEASURE1:60 for S being sigma_Field_Subset of X, M being sigma_Measure of S holds M is Measure of S; theorem :: MEASURE1:61 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; theorem :: MEASURE1:62 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; theorem :: MEASURE1:63 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; theorem :: MEASURE1:64 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; 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 :: MEASURE1:def 12 A in S; end; canceled; theorem :: MEASURE1:66 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; theorem :: MEASURE1:67 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; 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 :: MEASURE1:def 13 M.it = 0.; end; canceled; theorem :: MEASURE1:69 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; theorem :: MEASURE1:70 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; theorem :: MEASURE1:71 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;

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