Article MEASURE7, MML version 4.99.1005
:: MEASURE7:th 1
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
st for b2 being Element of NAT holds
b1 . b2 = 0.
holds SUM b1 = 0.;
:: MEASURE7:th 2
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
st b1 is nonnegative
for b2 being Element of NAT holds
b1 . b2 <= (Ser b1) . b2;
:: MEASURE7:th 3
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,ExtREAL
st b2 is nonnegative &
b3 is nonnegative &
(for b4 being Element of NAT holds
b1 . b4 = (b2 . b4) + (b3 . b4))
for b4 being Element of NAT holds
(Ser b1) . b4 = ((Ser b2) . b4) + ((Ser b3) . b4);
:: MEASURE7:th 4
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,ExtREAL
st (for b4 being Element of NAT holds
b1 . b4 = (b2 . b4) + (b3 . b4)) &
b2 is nonnegative &
b3 is nonnegative
holds SUM b1 <= (SUM b2) + SUM b3;
:: MEASURE7:th 6
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,ExtREAL
st b1 is nonnegative &
(for b3 being Element of NAT holds
b1 . b3 <= b2 . b3)
for b3 being Element of NAT holds
(Ser b1) . b3 <= SUM b2;
:: MEASURE7:th 7
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
st b1 is nonnegative
for b2 being Element of NAT holds
(Ser b1) . b2 <= SUM b1;
:: MEASURE7:funcnot 1 => MEASURE7:func 1
definition
let a1 be non empty set;
let a2 be Function-like quasi_total Relation of a1,ExtREAL;
func On A2 -> Function-like quasi_total Relation of NAT,ExtREAL means
for b1 being Element of NAT holds
(b1 in a1 implies it . b1 = a2 . b1) & (b1 in a1 or it . b1 = 0.);
end;
:: MEASURE7:def 1
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of b1,ExtREAL
for b3 being Function-like quasi_total Relation of NAT,ExtREAL holds
b3 = On b2
iff
for b4 being Element of NAT holds
(b4 in b1 implies b3 . b4 = b2 . b4) & (b4 in b1 or b3 . b4 = 0.);
:: MEASURE7:th 8
theorem
for b1 being non empty set
for b2 being Function-like quasi_total Relation of b1,ExtREAL
st b2 is nonnegative
holds On b2 is nonnegative;
:: MEASURE7:th 9
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
st b1 is nonnegative
for b2, b3 being Element of NAT
st b2 <= b3
holds (Ser b1) . b2 <= (Ser b1) . b3;
:: MEASURE7:th 10
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,ExtREAL
st for b3 being Element of NAT
st b3 <> b1
holds b2 . b3 = 0.
holds (for b3 being Element of NAT
st b3 < b1
holds (Ser b2) . b3 = 0.) &
(for b3 being Element of NAT
st b1 <= b3
holds (Ser b2) . b3 = b2 . b1);
:: MEASURE7:th 11
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
st b1 is nonnegative
for b2 being non empty Element of bool NAT
for b3 being Function-like quasi_total Relation of b2,NAT
st b3 is one-to-one
holds SUM On (b1 * b3) <= SUM b1;
:: MEASURE7:th 12
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,ExtREAL
st b1 is nonnegative & b2 is nonnegative
for b3 being non empty Element of bool NAT
for b4 being Function-like quasi_total Relation of b3,NAT
st b4 is one-to-one &
(for b5 being Element of NAT holds
(b5 in b3 implies b1 . b5 = (b2 * b4) . b5) &
(b5 in b3 or b1 . b5 = 0.))
holds SUM b1 <= SUM b2;
:: MEASURE7:modenot 1 => MEASURE7:mode 1
definition
let a1 be Element of bool REAL;
mode Interval_Covering of A1 -> Function-like quasi_total Relation of NAT,bool REAL means
a1 c= union rng it &
(for b1 being Element of NAT holds
it . b1 is interval Element of bool REAL);
end;
:: MEASURE7:dfs 2
definiens
let a1 be Element of bool REAL;
let a2 be Function-like quasi_total Relation of NAT,bool REAL;
To prove
a2 is Interval_Covering of a1
it is sufficient to prove
thus a1 c= union rng a2 &
(for b1 being Element of NAT holds
a2 . b1 is interval Element of bool REAL);
:: MEASURE7:def 2
theorem
for b1 being Element of bool REAL
for b2 being Function-like quasi_total Relation of NAT,bool REAL holds
b2 is Interval_Covering of b1
iff
b1 c= union rng b2 &
(for b3 being Element of NAT holds
b2 . b3 is interval Element of bool REAL);
:: MEASURE7:funcnot 2 => MEASURE7:func 2
definition
let a1 be Element of bool REAL;
let a2 be Interval_Covering of a1;
let a3 be Element of NAT;
redefine func a2 . a3 -> interval Element of bool REAL;
end;
:: MEASURE7:modenot 2 => MEASURE7:mode 2
definition
let a1 be Function-like quasi_total Relation of NAT,bool REAL;
mode Interval_Covering of A1 -> Function-like quasi_total Relation of NAT,Funcs(NAT,bool REAL) means
for b1 being Element of NAT holds
it . b1 is Interval_Covering of a1 . b1;
end;
:: MEASURE7:dfs 3
definiens
let a1 be Function-like quasi_total Relation of NAT,bool REAL;
let a2 be Function-like quasi_total Relation of NAT,Funcs(NAT,bool REAL);
To prove
a2 is Interval_Covering of a1
it is sufficient to prove
thus for b1 being Element of NAT holds
a2 . b1 is Interval_Covering of a1 . b1;
:: MEASURE7:def 3
theorem
for b1 being Function-like quasi_total Relation of NAT,bool REAL
for b2 being Function-like quasi_total Relation of NAT,Funcs(NAT,bool REAL) holds
b2 is Interval_Covering of b1
iff
for b3 being Element of NAT holds
b2 . b3 is Interval_Covering of b1 . b3;
:: MEASURE7:funcnot 3 => MEASURE7:func 3
definition
let a1 be Element of bool REAL;
let a2 be Interval_Covering of a1;
func A2 vol -> Function-like quasi_total Relation of NAT,ExtREAL means
for b1 being Element of NAT holds
it . b1 = vol (a2 . b1);
end;
:: MEASURE7:def 4
theorem
for b1 being Element of bool REAL
for b2 being Interval_Covering of b1
for b3 being Function-like quasi_total Relation of NAT,ExtREAL holds
b3 = b2 vol
iff
for b4 being Element of NAT holds
b3 . b4 = vol (b2 . b4);
:: MEASURE7:th 13
theorem
for b1 being Element of bool REAL
for b2 being Interval_Covering of b1 holds
b2 vol is nonnegative;
:: MEASURE7:funcnot 4 => MEASURE7:func 4
definition
let a1 be Function-like quasi_total Relation of NAT,bool REAL;
let a2 be Interval_Covering of a1;
let a3 be Element of NAT;
redefine func a2 . a3 -> Interval_Covering of a1 . a3;
end;
:: MEASURE7:funcnot 5 => MEASURE7:func 5
definition
let a1 be Function-like quasi_total Relation of NAT,bool REAL;
let a2 be Interval_Covering of a1;
func A2 vol -> Function-like quasi_total Relation of NAT,Funcs(NAT,ExtREAL) means
for b1 being Element of NAT holds
it . b1 = (a2 . b1) vol;
end;
:: MEASURE7:def 5
theorem
for b1 being Function-like quasi_total Relation of NAT,bool REAL
for b2 being Interval_Covering of b1
for b3 being Function-like quasi_total Relation of NAT,Funcs(NAT,ExtREAL) holds
b3 = b2 vol
iff
for b4 being Element of NAT holds
b3 . b4 = (b2 . b4) vol;
:: MEASURE7:funcnot 6 => MEASURE7:func 6
definition
let a1 be Element of bool REAL;
let a2 be Interval_Covering of a1;
func vol A2 -> Element of ExtREAL equals
SUM (a2 vol);
end;
:: MEASURE7:def 6
theorem
for b1 being Element of bool REAL
for b2 being Interval_Covering of b1 holds
vol b2 = SUM (b2 vol);
:: MEASURE7:funcnot 7 => MEASURE7:func 7
definition
let a1 be Function-like quasi_total Relation of NAT,bool REAL;
let a2 be Interval_Covering of a1;
func vol A2 -> Function-like quasi_total Relation of NAT,ExtREAL means
for b1 being Element of NAT holds
it . b1 = vol (a2 . b1);
end;
:: MEASURE7:def 7
theorem
for b1 being Function-like quasi_total Relation of NAT,bool REAL
for b2 being Interval_Covering of b1
for b3 being Function-like quasi_total Relation of NAT,ExtREAL holds
b3 = vol b2
iff
for b4 being Element of NAT holds
b3 . b4 = vol (b2 . b4);
:: MEASURE7:th 14
theorem
for b1 being Function-like quasi_total Relation of NAT,bool REAL
for b2 being Interval_Covering of b1
for b3 being Element of NAT holds
0. <= (vol b2) . b3;
:: MEASURE7:funcnot 8 => MEASURE7:func 8
definition
let a1 be Element of bool REAL;
func Svc A1 -> Element of bool ExtREAL means
for b1 being Element of ExtREAL holds
b1 in it
iff
ex b2 being Interval_Covering of a1 st
b1 = vol b2;
end;
:: MEASURE7:def 8
theorem
for b1 being Element of bool REAL
for b2 being Element of bool ExtREAL holds
b2 = Svc b1
iff
for b3 being Element of ExtREAL holds
b3 in b2
iff
ex b4 being Interval_Covering of b1 st
b3 = vol b4;
:: MEASURE7:funcreg 1
registration
let a1 be Element of bool REAL;
cluster Svc a1 -> non empty;
end;
:: MEASURE7:funcnot 9 => MEASURE7:func 9
definition
let a1 be Element of bool REAL;
func COMPLEX A1 -> Element of ExtREAL equals
inf Svc a1;
end;
:: MEASURE7:def 9
theorem
for b1 being Element of bool REAL holds
COMPLEX b1 = inf Svc b1;
:: MEASURE7:funcnot 10 => MEASURE7:func 10
definition
func OS_Meas -> Function-like quasi_total Relation of bool REAL,ExtREAL means
for b1 being Element of bool REAL holds
it . b1 = inf Svc b1;
end;
:: MEASURE7:def 10
theorem
for b1 being Function-like quasi_total Relation of bool REAL,ExtREAL holds
b1 = OS_Meas
iff
for b2 being Element of bool REAL holds
b1 . b2 = inf Svc b2;
:: MEASURE7:funcnot 11 => MEASURE7:func 11
definition
let a1 be Function-like quasi_total Relation of NAT,[:NAT,NAT:];
redefine func pr1 A1 -> Function-like quasi_total Relation of NAT,NAT means
for b1 being Element of NAT holds
ex b2 being Element of NAT st
a1 . b1 = [it . b1,b2];
end;
:: MEASURE7:def 11
theorem
for b1 being Function-like quasi_total Relation of NAT,[:NAT,NAT:]
for b2 being Function-like quasi_total Relation of NAT,NAT holds
b2 = pr1 b1
iff
for b3 being Element of NAT holds
ex b4 being Element of NAT st
b1 . b3 = [b2 . b3,b4];
:: MEASURE7:funcnot 12 => MEASURE7:func 12
definition
let a1 be Function-like quasi_total Relation of NAT,[:NAT,NAT:];
redefine func pr2 A1 -> Function-like quasi_total Relation of NAT,NAT means
for b1 being Element of NAT holds
a1 . b1 = [(pr1 a1) . b1,it . b1];
end;
:: MEASURE7:def 12
theorem
for b1 being Function-like quasi_total Relation of NAT,[:NAT,NAT:]
for b2 being Function-like quasi_total Relation of NAT,NAT holds
b2 = pr2 b1
iff
for b3 being Element of NAT holds
b1 . b3 = [(pr1 b1) . b3,b2 . b3];
:: MEASURE7:funcnot 13 => MEASURE7:func 13
definition
let a1 be Function-like quasi_total Relation of NAT,bool REAL;
let a2 be Interval_Covering of a1;
let a3 be Function-like quasi_total Relation of NAT,[:NAT,NAT:];
assume rng a3 = [:NAT,NAT:];
func On(A2,A3) -> Interval_Covering of union rng a1 means
for b1 being Element of NAT holds
it . b1 = (a2 . ((pr1 a3) . b1)) . ((pr2 a3) . b1);
end;
:: MEASURE7:def 13
theorem
for b1 being Function-like quasi_total Relation of NAT,bool REAL
for b2 being Interval_Covering of b1
for b3 being Function-like quasi_total Relation of NAT,[:NAT,NAT:]
st rng b3 = [:NAT,NAT:]
for b4 being Interval_Covering of union rng b1 holds
b4 = On(b2,b3)
iff
for b5 being Element of NAT holds
b4 . b5 = (b2 . ((pr1 b3) . b5)) . ((pr2 b3) . b5);
:: MEASURE7:th 15
theorem
for b1 being Function-like quasi_total Relation of NAT,[:NAT,NAT:]
st b1 is one-to-one & rng b1 = [:NAT,NAT:]
for b2 being Element of NAT holds
ex b3 being Element of NAT st
for b4 being Function-like quasi_total Relation of NAT,bool REAL
for b5 being Interval_Covering of b4 holds
(Ser ((On(b5,b1)) vol)) . b2 <= (Ser vol b5) . b3;
:: MEASURE7:th 16
theorem
for b1 being Function-like quasi_total Relation of NAT,bool REAL
for b2 being Interval_Covering of b1 holds
inf Svc union rng b1 <= SUM vol b2;
:: MEASURE7:th 17
theorem
OS_Meas is C_Measure of REAL;
:: MEASURE7:funcnot 14 => MEASURE7:func 14
definition
redefine func OS_Meas -> C_Measure of REAL;
end;
:: MEASURE7:funcnot 15 => MEASURE7:func 15
definition
func Lmi_sigmaFIELD -> non empty compl-closed sigma-multiplicative Element of bool bool REAL equals
sigma_Field OS_Meas;
end;
:: MEASURE7:def 14
theorem
Lmi_sigmaFIELD = sigma_Field OS_Meas;
:: MEASURE7:funcnot 16 => MEASURE7:func 16
definition
func L_mi -> sigma_Measure of Lmi_sigmaFIELD equals
sigma_Meas OS_Meas;
end;
:: MEASURE7:def 15
theorem
L_mi = sigma_Meas OS_Meas;