Article MEASURE3, MML version 4.99.1005
:: MEASURE3:th 1
theorem
for b1 being Element of ExtREAL
st -infty < b1 & b1 < +infty
holds b1 is Element of REAL;
:: MEASURE3:th 2
theorem
for b1 being ext-real set
st b1 <> -infty & b1 <> +infty
holds b1 is Element of REAL;
:: MEASURE3:th 3
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,ExtREAL
st for b3 being Element of NAT holds
(Ser b1) . b3 <= (Ser b2) . b3
holds SUM b1 <= SUM b2;
:: MEASURE3:th 4
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,ExtREAL
st for b3 being Element of NAT holds
(Ser b1) . b3 = (Ser b2) . b3
holds SUM b1 = SUM b2;
:: MEASURE3:modenot 1 => MEASURE2:mode 1
notation
let a1 be set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
synonym N_Sub_fam of a2 for N_Measure_fam of a2;
end;
:: MEASURE3:funcnot 1 => MEASURE3:func 1
definition
let a1 be set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be Function-like quasi_total Relation of NAT,a2;
redefine func rng a3 -> N_Measure_fam of a2;
end;
:: MEASURE3:th 5
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being Function-like quasi_total Relation of NAT,b2
for b5 being Element of b2
st meet rng b4 c= b5 &
(for b6 being Element of NAT holds
b5 c= b4 . b6)
holds b3 . b5 = b3 . meet rng b4;
:: MEASURE3:th 6
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Function-like quasi_total Relation of NAT,b2
st b3 . 0 = {} &
(for b5 being Element of NAT holds
b3 . (b5 + 1) = (b4 . 0) \ (b4 . b5) &
b4 . (b5 + 1) c= b4 . b5)
holds union rng b3 = (b4 . 0) \ meet rng b4;
:: MEASURE3:th 7
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Function-like quasi_total Relation of NAT,b2
st b3 . 0 = {} &
(for b5 being Element of NAT holds
b3 . (b5 + 1) = (b4 . 0) \ (b4 . b5) &
b4 . (b5 + 1) c= b4 . b5)
holds meet rng b4 = (b4 . 0) \ union rng b3;
:: MEASURE3:th 8
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4, b5 being Function-like quasi_total Relation of NAT,b2
st b3 . (b5 . 0) < +infty &
b4 . 0 = {} &
(for b6 being Element of NAT holds
b4 . (b6 + 1) = (b5 . 0) \ (b5 . b6) &
b5 . (b6 + 1) c= b5 . b6)
holds b3 . meet rng b5 = (b3 . (b5 . 0)) - (b3 . union rng b4);
:: MEASURE3:th 9
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4, b5 being Function-like quasi_total Relation of NAT,b2
st b3 . (b5 . 0) < +infty &
b4 . 0 = {} &
(for b6 being Element of NAT holds
b4 . (b6 + 1) = (b5 . 0) \ (b5 . b6) &
b5 . (b6 + 1) c= b5 . b6)
holds b3 . union rng b4 = (b3 . (b5 . 0)) - (b3 . meet rng b5);
:: MEASURE3:th 10
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4, b5 being Function-like quasi_total Relation of NAT,b2
st b3 . (b5 . 0) < +infty &
b4 . 0 = {} &
(for b6 being Element of NAT holds
b4 . (b6 + 1) = (b5 . 0) \ (b5 . b6) &
b5 . (b6 + 1) c= b5 . b6)
holds b3 . meet rng b5 = (b3 . (b5 . 0)) - sup rng (b3 * b4);
:: MEASURE3:th 11
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4, b5 being Function-like quasi_total Relation of NAT,b2
st b3 . (b5 . 0) < +infty &
b4 . 0 = {} &
(for b6 being Element of NAT holds
b4 . (b6 + 1) = (b5 . 0) \ (b5 . b6) &
b5 . (b6 + 1) c= b5 . b6)
holds sup rng (b3 * b4) is Element of REAL & b3 . (b5 . 0) is Element of REAL & inf rng (b3 * b5) is Element of REAL;
:: MEASURE3:th 12
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4, b5 being Function-like quasi_total Relation of NAT,b2
st b3 . (b5 . 0) < +infty &
b4 . 0 = {} &
(for b6 being Element of NAT holds
b4 . (b6 + 1) = (b5 . 0) \ (b5 . b6) &
b5 . (b6 + 1) c= b5 . b6)
holds sup rng (b3 * b4) = (b3 . (b5 . 0)) - inf rng (b3 * b5);
:: MEASURE3:th 13
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4, b5 being Function-like quasi_total Relation of NAT,b2
st b3 . (b5 . 0) < +infty &
b4 . 0 = {} &
(for b6 being Element of NAT holds
b4 . (b6 + 1) = (b5 . 0) \ (b5 . b6) &
b5 . (b6 + 1) c= b5 . b6)
holds inf rng (b3 * b5) = (b3 . (b5 . 0)) - sup rng (b3 * b4);
:: MEASURE3:th 14
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being Function-like quasi_total Relation of NAT,b2
st (for b5 being Element of NAT holds
b4 . (b5 + 1) c= b4 . b5) &
b3 . (b4 . 0) < +infty
holds b3 . meet rng b4 = inf rng (b3 * b4);
:: MEASURE3:th 15
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Measure of b2
for b4 being Function-like quasi_total disjoint_valued Relation of NAT,b2 holds
SUM (b3 * b4) <= b3 . union rng b4;
:: MEASURE3:th 17
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Measure of b2
st for b4 being Function-like quasi_total disjoint_valued Relation of NAT,b2 holds
b3 . union rng b4 <= SUM (b3 * b4)
holds b3 is sigma_Measure of b2;
:: MEASURE3:prednot 1 => MEASURE3:pred 1
definition
let a1 be set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be sigma_Measure of a2;
pred A3 is_complete A2 means
for b1 being Element of bool a1
for b2 being set
st b2 in a2 & b1 c= b2 & a3 . b2 = 0.
holds b1 in a2;
end;
:: MEASURE3:dfs 1
definiens
let a1 be set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be sigma_Measure of a2;
To prove
a3 is_complete a2
it is sufficient to prove
thus for b1 being Element of bool a1
for b2 being set
st b2 in a2 & b1 c= b2 & a3 . b2 = 0.
holds b1 in a2;
:: MEASURE3:def 2
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2 holds
b3 is_complete b2
iff
for b4 being Element of bool b1
for b5 being set
st b5 in b2 & b4 c= b5 & b3 . b5 = 0.
holds b4 in b2;
:: MEASURE3:modenot 2 => MEASURE3:mode 1
definition
let a1 be set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be sigma_Measure of a2;
mode thin of A3 -> Element of bool a1 means
ex b1 being set st
b1 in a2 & it c= b1 & a3 . b1 = 0.;
end;
:: MEASURE3:dfs 2
definiens
let a1 be set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be sigma_Measure of a2;
let a4 be Element of bool a1;
To prove
a4 is thin of a3
it is sufficient to prove
thus ex b1 being set st
b1 in a2 & a4 c= b1 & a3 . b1 = 0.;
:: MEASURE3:def 3
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being Element of bool b1 holds
b4 is thin of b3
iff
ex b5 being set st
b5 in b2 & b4 c= b5 & b3 . b5 = 0.;
:: MEASURE3:funcnot 2 => MEASURE3:func 2
definition
let a1 be set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be sigma_Measure of a2;
func COM(A2,A3) -> non empty Element of bool bool a1 means
for b1 being set holds
b1 in it
iff
ex b2 being set st
b2 in a2 &
(ex b3 being thin of a3 st
b1 = b2 \/ b3);
end;
:: MEASURE3:def 4
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being non empty Element of bool bool b1 holds
b4 = COM(b2,b3)
iff
for b5 being set holds
b5 in b4
iff
ex b6 being set st
b6 in b2 &
(ex b7 being thin of b3 st
b5 = b6 \/ b7);
:: MEASURE3:funcnot 3 => MEASURE3:func 3
definition
let a1 be set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be sigma_Measure of a2;
let a4 be Element of COM(a2,a3);
func MeasPart A4 -> non empty Element of bool bool a1 means
for b1 being set holds
b1 in it
iff
b1 in a2 & b1 c= a4 & a4 \ b1 is thin of a3;
end;
:: MEASURE3:def 5
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being Element of COM(b2,b3)
for b5 being non empty Element of bool bool b1 holds
b5 = MeasPart b4
iff
for b6 being set holds
b6 in b5
iff
b6 in b2 & b6 c= b4 & b4 \ b6 is thin of b3;
:: MEASURE3:th 18
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being Function-like quasi_total Relation of NAT,COM(b2,b3) holds
ex b5 being Function-like quasi_total Relation of NAT,b2 st
for b6 being Element of NAT holds
b5 . b6 in MeasPart (b4 . b6);
:: MEASURE3:th 19
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being Function-like quasi_total Relation of NAT,COM(b2,b3)
for b5 being Function-like quasi_total Relation of NAT,b2 holds
ex b6 being Function-like quasi_total Relation of NAT,bool b1 st
for b7 being Element of NAT holds
b6 . b7 = (b4 . b7) \ (b5 . b7);
:: MEASURE3:th 20
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being Function-like quasi_total Relation of NAT,bool b1
st for b5 being Element of NAT holds
b4 . b5 is thin of b3
holds ex b5 being Function-like quasi_total Relation of NAT,b2 st
for b6 being Element of NAT holds
b4 . b6 c= b5 . b6 & b3 . (b5 . b6) = 0.;
:: MEASURE3:th 21
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being non empty Element of bool bool b1
st for b5 being set holds
b5 in b4
iff
ex b6 being set st
b6 in b2 &
(ex b7 being thin of b3 st
b5 = b6 \/ b7)
holds b4 is non empty compl-closed sigma-multiplicative Element of bool bool b1;
:: MEASURE3:funcreg 1
registration
let a1 be set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be sigma_Measure of a2;
cluster COM(a2,a3) -> non empty compl-closed sigma-additive;
end;
:: MEASURE3:th 22
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4, b5 being set
st b4 in b2 & b5 in b2
for b6, b7 being thin of b3
st b4 \/ b6 = b5 \/ b7
holds b3 . b4 = b3 . b5;
:: MEASURE3:funcnot 4 => MEASURE3:func 4
definition
let a1 be set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be sigma_Measure of a2;
func COM A3 -> sigma_Measure of COM(a2,a3) means
for b1 being set
st b1 in a2
for b2 being thin of a3 holds
it . (b1 \/ b2) = a3 . b1;
end;
:: MEASURE3:def 6
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2
for b4 being sigma_Measure of COM(b2,b3) holds
b4 = COM b3
iff
for b5 being set
st b5 in b2
for b6 being thin of b3 holds
b4 . (b5 \/ b6) = b3 . b5;
:: MEASURE3:th 23
theorem
for b1 being set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being sigma_Measure of b2 holds
COM b3 is_complete COM(b2,b3);