Journal of Formalized Mathematics
Volume 12, 2000
University of Bialystok
Copyright (c) 2000 Association of Mizar Users

### Definitions and Basic Properties of Measurable Functions

by
Noboru Endou,
Katsumi Wasaki, and
Yasunari Shidama

MML identifier: MESFUNC1
[ Mizar article, MML identifier index ]

```environ

vocabulary INT_1, RAT_1, ARYTM_1, TARSKI, FUNCT_1, FUNCT_2, RELAT_1, BOOLE,
ARYTM_3, CARD_4, CARD_1, FINSET_1, PARTFUN1, SUPINF_1, SEQ_1, MEASURE6,
RLVECT_1, GROUP_1, COMPLEX1, MEASURE1, SETFAM_1, MESFUNC1, ARYTM;
notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0,
XREAL_0, MEASURE6, REAL_1, RELAT_1, FUNCT_1, NAT_1, INT_1, RAT_1,
FINSET_1, CARD_1, CARD_4, SUPINF_1, SUPINF_2, MEASURE1, MEASURE2,
MEASURE3, FUNCT_2, PARTFUN1, EXTREAL1;
constructors NAT_1, MEASURE3, MEASURE6, RAT_1, WELLORD2, EXTREAL1, REAL_1,
SUPINF_2, MEMBERED;
clusters SUBSET_1, SUPINF_1, MEASURE1, XREAL_0, RELSET_1, INT_1, CARD_1,
NAT_1, MEMBERED, NUMBERS, ORDINAL2;
requirements NUMERALS, BOOLE, SUBSET, REAL, ARITHM;

begin  :: Cardinal numbers of INT and RAT

reserve k for Nat;
reserve r for Real;
reserve i for Integer;
reserve q for Rational;

definition
func INT- -> Subset of REAL means
:: MESFUNC1:def 1
r in it iff ex k st r = - k;
end;

definition
cluster INT- -> non empty;
end;

theorem :: MESFUNC1:1
NAT,INT- are_equipotent;

theorem :: MESFUNC1:2
INT=INT- \/ NAT;

theorem :: MESFUNC1:3
NAT,INT are_equipotent;

definition
redefine func INT -> Subset of REAL;
end;

definition
let n be Nat;
func RAT_with_denominator n -> Subset of RAT means
:: MESFUNC1:def 2
q in it iff ex i st q = i/n;
end;

definition
let n be Nat;
cluster RAT_with_denominator(n+1) -> non empty;
end;

theorem :: MESFUNC1:4
for n being Nat holds INT,RAT_with_denominator (n+1) are_equipotent;

theorem :: MESFUNC1:5
NAT,RAT are_equipotent;

begin  :: Basic operations on R_EAL valued functions

definition let C be non empty set, f be PartFunc of C, ExtREAL, x be set;
redefine func f.x -> R_eal;
end;

definition let C be non empty set; let f1,f2 be PartFunc of C,ExtREAL;
func f1+f2 -> PartFunc of C,ExtREAL means
:: MESFUNC1:def 3
dom it =
(dom f1 /\ dom f2)\((f1"{-infty} /\ f2"{+infty}) \/ (f1"{+infty} /\
f2"{-infty}))
& for c being Element of C st c in dom it holds it.c = f1.c + f2.c;

func f1-f2 -> PartFunc of C,ExtREAL means
:: MESFUNC1:def 4
dom it =
(dom f1 /\ dom f2)\((f1"{+infty} /\ f2"{+infty}) \/ (f1"{-infty} /\
f2"{-infty}))
& for c being Element of C st c in dom it holds it.c = f1.c - f2.c;

func f1(#)f2 -> PartFunc of C,ExtREAL means
:: MESFUNC1:def 5
dom it = dom f1 /\ dom f2
& for c being Element of C st c in dom it holds it.c = f1.c * f2.c;
end;

definition let C be non empty set, f be PartFunc of C,ExtREAL, r be Real;
func r(#)f -> PartFunc of C,ExtREAL means
:: MESFUNC1:def 6
dom it = dom f
& for c being Element of C st c in dom it holds it.c = (R_EAL r) * f.c;
end;

theorem :: MESFUNC1:6
for C being non empty set, f being PartFunc of C,ExtREAL, r being Real
st r <> 0 holds
for c being Element of C st c in dom(r(#)f) holds f.c = (r(#)f).c / R_EAL r;

definition let C be non empty set; let f be PartFunc of C,ExtREAL;
func -f -> PartFunc of C,ExtREAL means
:: MESFUNC1:def 7
dom it = dom f & for c being Element of C st c in dom it holds it.c = -(f.c);
end;

definition
func 1. -> R_eal equals
:: MESFUNC1:def 8
1;
end;

definition let C be non empty set; let f be PartFunc of C,ExtREAL;
let r be Real;
func r/f -> PartFunc of C,ExtREAL means
:: MESFUNC1:def 9
dom it = dom f \ f"{0.} &
for c being Element of C st c in dom it holds it.c = (R_EAL r)/(f.c);
end;

theorem :: MESFUNC1:7
for C being non empty set, f being PartFunc of C,ExtREAL holds
dom (1/f) = dom f \ f"{0.} &
for c being Element of C st c in dom (1/f) holds (1/f).c = 1./(f.c);

definition let C be non empty set; let f be PartFunc of C,ExtREAL;
func |.f.| -> PartFunc of C,ExtREAL means
:: MESFUNC1:def 10
dom it = dom f &
for c being Element of C st c in dom it holds it.c = |. f.c .|;
end;

canceled;

theorem :: MESFUNC1:9
for C being non empty set, f1,f2 being PartFunc of C,ExtREAL holds
f1 + f2 = f2 + f1;

theorem :: MESFUNC1:10
for C being non empty set, f1,f2 being PartFunc of C,ExtREAL holds
f1 (#) f2 = f2 (#) f1;

definition let C be non empty set; let f1,f2 be PartFunc of C,ExtREAL;
redefine func f1+f2;
commutativity;

redefine func f1(#)f2;
commutativity;
end;

begin  :: Level sets

theorem :: MESFUNC1:11
for r being Real holds ex n being Nat st r <= n;

theorem :: MESFUNC1:12
for r being Real holds ex n being Nat st -n <= r;

theorem :: MESFUNC1:13
for r,s being real number st r < s holds
ex n being Nat st 1/(n+1) < s-r;

theorem :: MESFUNC1:14
for r,s being real number st for n being Nat holds r-1/(n+1) <= s holds r <= s;

theorem :: MESFUNC1:15
for a being R_eal st (for r being Real holds R_EAL r <' a) holds a = +infty;

theorem :: MESFUNC1:16
for a being R_eal st (for r being Real holds a <' R_EAL r) holds a = -infty;

definition
let X be set;
let S be sigma_Field_Subset of X;
let A be set;
pred A is_measurable_on S means
:: MESFUNC1:def 11
A in S;
end;

theorem :: MESFUNC1:17
for X,A being set, S being sigma_Field_Subset of X holds
A is_measurable_on S iff
(for M being sigma_Measure of S holds A is_measurable M);

reserve X for non empty set;
reserve x for Element of X;
reserve f,g for PartFunc of X,ExtREAL;
reserve S for sigma_Field_Subset of X;
reserve F for Function of NAT,S;
reserve A for set;
reserve a for R_eal;
reserve r,s for Real;
reserve n,m for Nat;

definition
let X,f,a;
func less_dom(f,a) -> Subset of X means
:: MESFUNC1:def 12
x in it iff x in dom f & ex y being R_eal st y=f.x & y <' a;

func less_eq_dom(f,a) -> Subset of X means
:: MESFUNC1:def 13
x in it iff x in dom f & ex y being R_eal st y=f.x & y <=' a;

func great_dom(f,a) -> Subset of X means
:: MESFUNC1:def 14
x in it iff x in dom f & ex y being R_eal st y=f.x & a <' y;

func great_eq_dom(f,a) -> Subset of X means
:: MESFUNC1:def 15
x in it iff x in dom f & ex y being R_eal st y=f.x & a <=' y;

func eq_dom(f,a) -> Subset of X means
:: MESFUNC1:def 16
x in it iff x in dom f & ex y being R_eal st y=f.x & a=y;
end;

theorem :: MESFUNC1:18
for X, S, f, A, a st A c= dom f holds
A /\ great_eq_dom(f,a) = A\(A /\ less_dom(f,a));

theorem :: MESFUNC1:19
for X, S, f, A, a st A c= dom f holds
A /\ great_dom(f,a) = A\(A /\ less_eq_dom(f,a));

theorem :: MESFUNC1:20
for X, S, f, A, a st A c= dom f holds
A /\ less_eq_dom(f,a) = A\(A /\ great_dom(f,a));

theorem :: MESFUNC1:21
for X, S, f, A, a st A c= dom f holds
A /\ less_dom(f,a) = A\(A /\ great_eq_dom(f,a));

theorem :: MESFUNC1:22
for X, S, f, A, a holds
A /\ eq_dom(f,a) = A /\ great_eq_dom(f,a) /\ less_eq_dom(f,a);

theorem :: MESFUNC1:23
for X, S, F, f, A, r st for n holds F.n = A /\ great_dom(f,R_EAL(r-1/(n+1)))
holds A /\ great_eq_dom(f,R_EAL r) = meet rng F;

theorem :: MESFUNC1:24
for X, S, F, f, A
for r being real number st for n holds F.n = A /\ less_dom(f,R_EAL(r+1/(n+1)))
holds A /\ less_eq_dom(f,R_EAL r) = meet rng F;

theorem :: MESFUNC1:25
for X, S, F, f, A
for r being real number st for n
holds F.n = A /\ less_eq_dom(f,R_EAL(r-1/(n+1)))
holds A /\ less_dom(f,R_EAL r) = union rng F;

theorem :: MESFUNC1:26
for X, S, F, f, A, r st for n holds F.n = A /\
great_eq_dom(f,R_EAL(r+1/(n+1)))
holds A /\ great_dom(f,R_EAL r) = union rng F;

theorem :: MESFUNC1:27
for X, S, F, f, A st for n holds F.n = A /\ great_dom(f,R_EAL n)
holds A /\ eq_dom(f,+infty) = meet rng F;

theorem :: MESFUNC1:28
for X, S, F, f, A st for n holds F.n = A /\ less_dom(f,R_EAL n)
holds A /\ less_dom(f,+infty) = union rng F;

theorem :: MESFUNC1:29
for X, S, F, f, A st for n holds F.n = A /\ less_dom(f,R_EAL (-n))
holds A /\ eq_dom(f,-infty) = meet rng F;

theorem :: MESFUNC1:30
for X, S, F, f, A st for n holds F.n = A /\ great_dom(f,R_EAL (-n))
holds A /\ great_dom(f,-infty) = union rng F;

begin :: Measurable functions

definition
let X be non empty set;
let S be sigma_Field_Subset of X;
let f be PartFunc of X,ExtREAL;
let A be Element of S;
pred f is_measurable_on A means
:: MESFUNC1:def 17
for r being real number holds A /\ less_dom(f,R_EAL r) is_measurable_on S;
end;

reserve A,B for Element of S;

theorem :: MESFUNC1:31
for X,S,f,A st A c= dom f holds f is_measurable_on A iff
(for r being real number holds A /\ great_eq_dom(f,R_EAL r) is_measurable_on S)
;

theorem :: MESFUNC1:32
for X,S,f,A holds f is_measurable_on A iff
(for r being real number holds A /\ less_eq_dom(f,R_EAL r) is_measurable_on S);

theorem :: MESFUNC1:33
for X,S,f,A st A c= dom f holds f is_measurable_on A iff
(for r being real number holds A /\ great_dom(f,R_EAL r) is_measurable_on S);

theorem :: MESFUNC1:34
for X,S,f,A,B st B c= A & f is_measurable_on A holds f is_measurable_on B;

theorem :: MESFUNC1:35
for X,S,f,A,B st f is_measurable_on A & f is_measurable_on B holds
f is_measurable_on (A \/ B);

theorem :: MESFUNC1:36
for X,S,f,A,r,s st f is_measurable_on A & A c= dom f holds
(A /\ great_dom(f,R_EAL r) /\ less_dom(f,R_EAL s)) is_measurable_on S;

theorem :: MESFUNC1:37
for X,S,f,A st f is_measurable_on A & A c= dom f holds
A /\ eq_dom(f,+infty) is_measurable_on S;

theorem :: MESFUNC1:38
for X,S,f,A st f is_measurable_on A holds
A /\ eq_dom(f,-infty) is_measurable_on S;

theorem :: MESFUNC1:39
for X,S,f,A st f is_measurable_on A & A c= dom f holds
A /\ great_dom(f,-infty) /\ less_dom(f,+infty) is_measurable_on S;

theorem :: MESFUNC1:40
for X,S,f,g,A,r st f is_measurable_on A & g is_measurable_on A & A c= dom g
holds A /\ less_dom(f,R_EAL r) /\ great_dom(g,R_EAL r) is_measurable_on S;

theorem :: MESFUNC1:41
for X,S,f,A,r st f is_measurable_on A & A c= dom f
holds r(#)f is_measurable_on A;
```