Article MESFUNC2, MML version 4.99.1005
:: MESFUNC2:attrnot 1 => VALUED_0:attr 3
notation
let a1 be non empty set;
let a2 be Function-like Relation of a1,ExtREAL;
synonym finite for real-valued;
end;
:: MESFUNC2:attrnot 2 => MESFUNC2:attr 1
definition
let a1 be non empty set;
let a2 be Function-like Relation of a1,ExtREAL;
redefine attr a2 is finite means
for b1 being Element of a1
st b1 in dom a2
holds |.a2 . b1.| < +infty;
end;
:: MESFUNC2:dfs 1
definiens
let a1 be non empty set;
let a2 be Function-like Relation of a1,ExtREAL;
To prove
a1 is real-valued
it is sufficient to prove
thus for b1 being Element of a1
st b1 in dom a2
holds |.a2 . b1.| < +infty;
:: MESFUNC2:def 1
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL holds
b2 is real-valued
iff
for b3 being Element of b1
st b3 in dom b2
holds |.b2 . b3.| < +infty;
:: MESFUNC2:th 1
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL holds
b2 = 1 (#) b2;
:: MESFUNC2:th 2
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,ExtREAL
st (b2 is real-valued or b3 is real-valued)
holds dom (b2 + b3) = (dom b2) /\ dom b3 &
dom (b2 - b3) = (dom b2) /\ dom b3;
:: MESFUNC2:th 3
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Function-like Relation of b1,ExtREAL
for b5 being Function-like quasi_total Relation of RAT,b2
for b6 being Element of REAL
for b7 being Element of b2
st b3 is real-valued &
b4 is real-valued &
(for b8 being rational set holds
b5 . b8 = (b7 /\ less_dom(b3,R_EAL b8)) /\ (b7 /\ less_dom(b4,R_EAL (b6 - b8))))
holds b7 /\ less_dom(b3 + b4,R_EAL b6) = union rng b5;
:: MESFUNC2:th 4
theorem
ex b1 being Function-like quasi_total Relation of NAT,RAT st
b1 is one-to-one & dom b1 = NAT & rng b1 = RAT;
:: MESFUNC2:th 5
theorem
for b1, b2, b3 being non empty set
for b4 being Function-like quasi_total Relation of b1,b3
st b1,b2 are_equipotent
holds ex b5 being Function-like quasi_total Relation of b2,b3 st
rng b4 = rng b5;
:: MESFUNC2:th 6
theorem
for b1 being non empty set
for b2 being Element of REAL
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b4, b5 being Function-like Relation of b1,ExtREAL
for b6 being Element of b3
st b4 is_measurable_on b6 & b5 is_measurable_on b6
holds ex b7 being Function-like quasi_total Relation of RAT,b3 st
for b8 being rational set holds
b7 . b8 = (b6 /\ less_dom(b4,R_EAL b8)) /\ (b6 /\ less_dom(b5,R_EAL (b2 - b8)));
:: MESFUNC2:th 7
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Function-like Relation of b1,ExtREAL
for b5 being Element of b2
st b3 is real-valued & b4 is real-valued & b3 is_measurable_on b5 & b4 is_measurable_on b5
holds b3 + b4 is_measurable_on b5;
:: MESFUNC2:th 9
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,ExtREAL holds
b2 - b3 = b2 + - b3;
:: MESFUNC2:th 11
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL holds
- b2 = (- 1) (#) b2;
:: MESFUNC2:th 12
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being Element of REAL
st b2 is real-valued
holds b3 (#) b2 is real-valued;
:: MESFUNC2:th 13
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Function-like Relation of b1,ExtREAL
for b5 being Element of b2
st b3 is real-valued & b4 is real-valued & b3 is_measurable_on b5 & b4 is_measurable_on b5 & b5 c= dom b4
holds b3 - b4 is_measurable_on b5;
:: MESFUNC2:funcnot 1 => MESFUNC2:func 1
definition
let a1 be non empty set;
let a2 be Function-like Relation of a1,ExtREAL;
func max+ A2 -> Function-like Relation of a1,ExtREAL means
dom it = dom a2 &
(for b1 being Element of a1
st b1 in dom it
holds it . b1 = max(a2 . b1,0.));
end;
:: MESFUNC2:def 2
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,ExtREAL holds
b3 = max+ b2
iff
dom b3 = dom b2 &
(for b4 being Element of b1
st b4 in dom b3
holds b3 . b4 = max(b2 . b4,0.));
:: MESFUNC2:funcnot 2 => MESFUNC2:func 2
definition
let a1 be non empty set;
let a2 be Function-like Relation of a1,ExtREAL;
func max- A2 -> Function-like Relation of a1,ExtREAL means
dom it = dom a2 &
(for b1 being Element of a1
st b1 in dom it
holds it . b1 = max(- (a2 . b1),0.));
end;
:: MESFUNC2:def 3
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,ExtREAL holds
b3 = max- b2
iff
dom b3 = dom b2 &
(for b4 being Element of b1
st b4 in dom b3
holds b3 . b4 = max(- (b2 . b4),0.));
:: MESFUNC2:th 14
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being Element of b1 holds
0. <= (max+ b2) . b3;
:: MESFUNC2:th 15
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being Element of b1 holds
0. <= (max- b2) . b3;
:: MESFUNC2:th 16
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL holds
max- b2 = max+ - b2;
:: MESFUNC2:th 17
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being Element of b1
st 0. < (max+ b2) . b3
holds (max- b2) . b3 = 0.;
:: MESFUNC2:th 18
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being Element of b1
st 0. < (max- b2) . b3
holds (max+ b2) . b3 = 0.;
:: MESFUNC2:th 19
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL holds
dom b2 = dom ((max+ b2) - max- b2) &
dom b2 = dom ((max+ b2) + max- b2);
:: MESFUNC2:th 20
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being Element of b1 holds
((max+ b2) . b3 = b2 . b3 or (max+ b2) . b3 = 0.) &
((max- b2) . b3 = - (b2 . b3) or (max- b2) . b3 = 0.);
:: MESFUNC2:th 21
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being Element of b1
st (max+ b2) . b3 = b2 . b3
holds (max- b2) . b3 = 0.;
:: MESFUNC2:th 22
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being Element of b1
st b3 in dom b2 & (max+ b2) . b3 = 0.
holds (max- b2) . b3 = - (b2 . b3);
:: MESFUNC2:th 23
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being Element of b1
st (max- b2) . b3 = - (b2 . b3)
holds (max+ b2) . b3 = 0.;
:: MESFUNC2:th 24
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being Element of b1
st b3 in dom b2 & (max- b2) . b3 = 0.
holds (max+ b2) . b3 = b2 . b3;
:: MESFUNC2:th 25
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL holds
b2 = (max+ b2) - max- b2;
:: MESFUNC2:th 26
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL holds
|.b2.| = (max+ b2) + max- b2;
:: MESFUNC2:th 27
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b4 being Element of b3
st b2 is_measurable_on b4
holds max+ b2 is_measurable_on b4;
:: MESFUNC2:th 28
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b4 being Element of b3
st b2 is_measurable_on b4 & b4 c= dom b2
holds max- b2 is_measurable_on b4;
:: MESFUNC2:th 29
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like Relation of b1,ExtREAL
for b4 being Element of b2
st b3 is_measurable_on b4 & b4 c= dom b3
holds |.b3.| is_measurable_on b4;
:: MESFUNC2:funcnot 3 => MESFUNC2:func 3
definition
let a1, a2 be set;
redefine func chi(a1,a2) -> Function-like Relation of a2,ExtREAL;
end;
:: MESFUNC2:th 31
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Element of b2 holds
chi(b3,b1) is real-valued;
:: MESFUNC2:th 32
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3, b4 being Element of b2 holds
chi(b3,b1) is_measurable_on b4;
:: MESFUNC2:exreg 1
registration
let a1 be set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
cluster Relation-like Function-like finite FinSequence-like disjoint_valued FinSequence of a2;
end;
:: MESFUNC2:modenot 1
definition
let a1 be set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
mode Finite_Sep_Sequence of a2 is disjoint_valued FinSequence of a2;
end;
:: MESFUNC2:th 33
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Relation-like Function-like set
st b3 is disjoint_valued FinSequence of b2
holds ex b4 being Function-like quasi_total disjoint_valued Relation of NAT,b2 st
union proj2 b3 = union rng b4 &
(for b5 being natural set
st b5 in proj1 b3
holds b3 . b5 = b4 . b5) &
(for b5 being natural set
st not b5 in proj1 b3
holds b4 . b5 = {});
:: MESFUNC2:th 34
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Relation-like Function-like set
st b3 is disjoint_valued FinSequence of b2
holds union proj2 b3 in b2;
:: MESFUNC2:prednot 1 => MESFUNC2:pred 1
definition
let a1 be non empty set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be Function-like Relation of a1,ExtREAL;
pred A3 is_simple_func_in A2 means
a3 is real-valued &
(ex b1 being disjoint_valued FinSequence of a2 st
dom a3 = union rng b1 &
(for b2 being natural set
for b3, b4 being Element of a1
st b2 in dom b1 & b3 in b1 . b2 & b4 in b1 . b2
holds a3 . b3 = a3 . b4));
end;
:: MESFUNC2:dfs 4
definiens
let a1 be non empty set;
let a2 be non empty compl-closed sigma-multiplicative Element of bool bool a1;
let a3 be Function-like Relation of a1,ExtREAL;
To prove
a3 is_simple_func_in a2
it is sufficient to prove
thus a3 is real-valued &
(ex b1 being disjoint_valued FinSequence of a2 st
dom a3 = union rng b1 &
(for b2 being natural set
for b3, b4 being Element of a1
st b2 in dom b1 & b3 in b1 . b2 & b4 in b1 . b2
holds a3 . b3 = a3 . b4));
:: MESFUNC2:def 5
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being Function-like Relation of b1,ExtREAL holds
b3 is_simple_func_in b2
iff
b3 is real-valued &
(ex b4 being disjoint_valued FinSequence of b2 st
dom b3 = union rng b4 &
(for b5 being natural set
for b6, b7 being Element of b1
st b5 in dom b4 & b6 in b4 . b5 & b7 in b4 . b5
holds b3 . b6 = b3 . b7));
:: MESFUNC2:th 35
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
st b2 is real-valued
holds rng b2 is Element of bool REAL;
:: MESFUNC2:th 36
theorem
for b1 being non empty set
for b2 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b3 being natural set
for b4 being Relation-like set
st b4 is disjoint_valued FinSequence of b2
holds b4 | Seg b3 is disjoint_valued FinSequence of b2;
:: MESFUNC2:th 37
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,ExtREAL
for b3 being non empty compl-closed sigma-multiplicative Element of bool bool b1
for b4 being Element of b3
st b2 is_simple_func_in b3
holds b2 is_measurable_on b4;