begin
theorem Th1:
theorem Th2:
theorem Th3:
:: deftheorem Def1 defines FinSequence MEASURE8:def 1 :
:: deftheorem Def2 defines Set_Sequence MEASURE8:def 2 :
:: deftheorem Def3 defines Covering MEASURE8:def 3 :
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 :
:: deftheorem Def5 defines vol MEASURE8:def 5 :
theorem Th4:
:: deftheorem Def6 defines Volume MEASURE8:def 6 :
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 :
:: deftheorem Def8 defines C_Meas MEASURE8:def 8 :
:: deftheorem defines InvPairFunc MEASURE8:def 9 :
:: deftheorem Def10 defines On MEASURE8:def 10 :
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 :
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 :
theorem
theorem Th33:
theorem Th34:
theorem