begin
theorem Th1:
theorem Th2:
theorem Th3:
:: deftheorem Def1 defines FinSequence MEASURE8:def 1 :
for X being set
for F being Field_Subset of X
for b3 being FinSequence of bool X holds
( b3 is FinSequence of F iff for k being Nat st k in dom b3 holds
b3 . k in F );
:: deftheorem Def2 defines Set_Sequence MEASURE8:def 2 :
for X being set
for F being Field_Subset of X
for b3 being SetSequence of X holds
( b3 is Set_Sequence of F iff for n being Nat holds b3 . n in F );
:: deftheorem Def3 defines Covering MEASURE8:def 3 :
for X being set
for A being Subset of X
for F being Field_Subset of X
for b4 being Set_Sequence of F holds
( b4 is Covering of A,F iff A c= union (rng b4) );
Lm1:
for X being set
for F being Field_Subset of X holds NAT --> X is Set_Sequence of F
:: deftheorem Def4 defines Covering MEASURE8:def 4 :
for X being set
for F being Field_Subset of X
for Sets being SetSequence of X
for b4 being Function of NAT,(Funcs (NAT,(bool X))) holds
( b4 is Covering of Sets,F iff for n being Nat holds b4 . n is Covering of Sets . n,F );
:: deftheorem Def5 defines vol MEASURE8:def 5 :
for X being set
for F being Field_Subset of X
for M being Measure of F
for FSets being Set_Sequence of F
for b5 being ExtREAL_sequence holds
( b5 = vol (M,FSets) iff for n being natural number holds b5 . n = M . (FSets . n) );
theorem Th4:
:: deftheorem Def6 defines Volume MEASURE8:def 6 :
for X being set
for F being Field_Subset of X
for Sets being SetSequence of X
for M being Measure of F
for Cvr being Covering of Sets,F
for b6 being ExtREAL_sequence holds
( b6 = Volume (M,Cvr) iff for n being Nat holds b6 . n = SUM (vol (M,(Cvr . n))) );
theorem Th5:
definition
let X be
set ;
let F be
Field_Subset of
X;
let M be
Measure of
F;
let A be
Subset of
X;
func Svc (
M,
A)
-> Subset of
ExtREAL means :
Def7:
for
x being
R_eal holds
(
x in it iff ex
CA being
Covering of
A,
F st
x = SUM (vol (M,CA)) );
existence
ex b1 being Subset of ExtREAL st
for x being R_eal holds
( x in b1 iff ex CA being Covering of A,F st x = SUM (vol (M,CA)) )
uniqueness
for b1, b2 being Subset of ExtREAL st ( for x being R_eal holds
( x in b1 iff ex CA being Covering of A,F st x = SUM (vol (M,CA)) ) ) & ( for x being R_eal holds
( x in b2 iff ex CA being Covering of A,F st x = SUM (vol (M,CA)) ) ) holds
b1 = b2
end;
:: deftheorem Def7 defines Svc MEASURE8:def 7 :
for X being set
for F being Field_Subset of X
for M being Measure of F
for A being Subset of X
for b5 being Subset of ExtREAL holds
( b5 = Svc (M,A) iff for x being R_eal holds
( x in b5 iff ex CA being Covering of A,F st x = SUM (vol (M,CA)) ) );
:: deftheorem Def8 defines C_Meas MEASURE8:def 8 :
for X being set
for F being Field_Subset of X
for M being Measure of F
for b4 being Function of (bool X),ExtREAL holds
( b4 = C_Meas M iff for A being Subset of X holds b4 . A = inf (Svc (M,A)) );
:: deftheorem defines InvPairFunc MEASURE8:def 9 :
InvPairFunc = PairFunc " ;
:: deftheorem Def10 defines On MEASURE8:def 10 :
for X being set
for F being Field_Subset of X
for Sets being SetSequence of X
for Cvr being Covering of Sets,F
for b5 being Covering of union (rng Sets),F holds
( b5 = On Cvr iff for n being Nat holds b5 . n = (Cvr . ((pr1 InvPairFunc) . n)) . ((pr2 InvPairFunc) . n) );
theorem Th6:
theorem Th7:
theorem Th8:
theorem Th9:
theorem Th10:
theorem Th11:
theorem Th12:
Lm2:
for X being set
for F being Field_Subset of X
for M being Measure of F
for Sets being SetSequence of X st ex n being Nat st (C_Meas M) . (Sets . n) = +infty holds
(C_Meas M) . (union (rng Sets)) <= SUM ((C_Meas M) * Sets)
theorem Th13:
theorem Th14:
begin
:: deftheorem Def11 defines completely-additive MEASURE8:def 11 :
for X being set
for F being Field_Subset of X
for M being Measure of F holds
( M is completely-additive iff for FSets being Sep_Sequence of F st union (rng FSets) in F holds
SUM (M * FSets) = M . (union (rng FSets)) );
theorem Th15:
theorem Th16:
theorem Th17:
theorem Th18:
theorem Th19:
theorem Th20:
theorem Th21:
theorem Th22:
theorem Th23:
theorem Th24:
theorem
theorem Th26:
theorem
theorem
theorem Th29:
theorem Th30:
theorem
:: deftheorem Def12 defines is_extension_of MEASURE8:def 12 :
for X being set
for F being Field_Subset of X
for S being SigmaField of X
for m being Measure of F
for M being sigma_Measure of S holds
( M is_extension_of m iff for A being set st A in F holds
M . A = m . A );
theorem
theorem Th33:
theorem Th34:
theorem