MESFUNC1: Definitions and Basic Properties of Measurable Functions

:: Definitions and Basic Properties of Measurable Functions
:: by Noboru Endou , Katsumi Wasaki and Yasunari Shidama
::
:: Received September 7, 2000
:: Copyright (c) 2000-2024 Association of Mizar Users

definition

func INT- -> Subset of REAL means :Def1: :: MESFUNC1:def 1
for r being Real holds
( r in it iff ex k being Element of NAT st r = - k );

existence

proof end;

uniqueness

proof end;

correctness ;

end;

:: deftheorem Def1 defines INT- MESFUNC1:def 1 :

Lm1: 0 = - 0
;

registration

cluster INT- -> non empty ;

coherence by Def1, Lm1;

end;

theorem Th1: :: MESFUNC1:1

NAT , INT- are_equipotent

proof end;

theorem Th2: :: MESFUNC1:2

INT = INT- \/ NAT

proof end;

theorem Th3: :: MESFUNC1:3

NAT , INT are_equipotent by Th1, Th2, CARD_2:77;

definition

let n be Nat;

func RAT_with_denominator n -> Subset of RAT means :Def2: :: MESFUNC1:def 2
for q being Rational holds
( q in it iff ex i being Integer st q = i / n );

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def2 defines RAT_with_denominator MESFUNC1:def 2 :

registration

let n be Nat;

cluster RAT_with_denominator (n + 1) -> non empty ;

coherence

proof end;

end;

theorem Th4: :: MESFUNC1:4

for n being Nat holds INT , RAT_with_denominator (n + 1) are_equipotent

proof end;

theorem :: MESFUNC1:5

NAT , RAT are_equipotent

proof end;

definition

let C be non empty set ;
let f1, f2 be C -defined ExtREAL -valued Function;
deffunc H₁( Element of C) -> Element of ExtREAL = (f1 . $1) + (f2 . $1);

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) ) );

existence

proof end;

uniqueness

proof end;

commutativity ;
deffunc H₂( Element of C) -> Element of ExtREAL = (f1 . $1) - (f2 . $1);

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) ) );

existence

proof end;

uniqueness

proof end;

deffunc H₃( Element of C) -> Element of ExtREAL = (f1 . $1) * (f2 . $1);

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) ) );

existence

proof end;

uniqueness

proof end;

commutativity ;

end;

:: deftheorem defines + MESFUNC1:def 3 :

:: deftheorem defines - MESFUNC1:def 4 :

:: deftheorem defines (#) MESFUNC1:def 5 :

definition

let C be non empty set ;
let f be C -defined ExtREAL -valued Function;
let r be Real;
deffunc H₁( Element of C) -> Element of ExtREAL = In ((r * (f . $1)),ExtREAL);

func r (#) f -> PartFunc of C,ExtREAL means :Def6: :: MESFUNC1:def 6
( dom it = dom f & ( for c being Element of C st c in dom it holds
it . c = r * (f . c) ) );

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def6 defines (#) MESFUNC1:def 6 :

theorem Th6: :: MESFUNC1:6

for C being non empty set
for f being PartFunc of C,ExtREAL
for 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

proof end;

definition

let C be non empty set ;
let f be C -defined ExtREAL -valued Function;
deffunc H₁( Element of C) -> Element of ExtREAL = - (f . $1);

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) ) );

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem defines - MESFUNC1:def 7 :

canceled;

::$CD

definition

let C be non empty set ;
let f be C -defined ExtREAL -valued Function;
let r be Real;
deffunc H₁( Element of C) -> Element of ExtREAL = In ((r / (f . $1)),ExtREAL);

func r / f -> PartFunc of C,ExtREAL means :Def9: :: MESFUNC1:def 8
( dom it = (dom f) \ (f " {0.}) & ( for c being Element of C st c in dom it holds
it . c = r / (f . c) ) );

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def9 defines / MESFUNC1:def 8 :

theorem :: MESFUNC1:7

for C being non empty set
for 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) ) ) by Def9;

definition

let C be non empty set ;
let f be C -defined ExtREAL -valued Function;
deffunc H₁( Element of C) -> Element of ExtREAL = |.(f . $1).|;

func |.f.| -> PartFunc of C,ExtREAL means :: MESFUNC1:def 9
( dom it = dom f & ( for c being Element of C st c in dom it holds
it . c = |.(f . c).| ) );

existence

proof end;

uniqueness

proof end;

end;

:: deftheorem defines |. MESFUNC1:def 9 :

theorem Th8: :: MESFUNC1:8

for r being Real ex n being Element of NAT st r <= n

proof end;

theorem Th9: :: MESFUNC1:9

for r being Real ex n being Nat st - n <= r

proof end;

theorem Th10: :: MESFUNC1:10

for r, s being Real st r < s holds
ex n being Element of NAT st 1 / (n + 1) < s - r

proof end;

theorem Th11: :: MESFUNC1:11

for r, s being Real st ( for n being Element of NAT holds r - (1 / (n + 1)) <= s ) holds
r <= s

proof end;

theorem Th12: :: MESFUNC1:12

for a being R_eal st ( for r being Real holds r < a ) holds
a = +infty

proof end;

theorem Th13: :: MESFUNC1:13

for a being R_eal st ( for r being Real holds a < r ) holds
a = -infty

proof end;

notation

let f be ext-real-valued Function;
let a be ExtReal;
synonym eq_dom (f,a) for Coim (f,a);

end;

definition

let f be ext-real-valued Function;
let a be ExtReal;
defpred S₁[ object ] means ( $1 in dom f & f . $1 < a );

func less_dom (f,a) -> set means :Def11: :: MESFUNC1:def 10
for x being object holds
( x in it iff ( x in dom f & f . x < a ) );

existence

proof end;

uniqueness

proof end;

defpred S₂[ object ] means ( $1 in dom f & f . $1 <= a );

func less_eq_dom (f,a) -> set means :Def12: :: MESFUNC1:def 11
for x being object holds
( x in it iff ( x in dom f & f . x <= a ) );

existence

proof end;

uniqueness

proof end;

defpred S₃[ object ] means ( $1 in dom f & a < f . $1 );

func great_dom (f,a) -> set means :Def13: :: MESFUNC1:def 12
for x being object holds
( x in it iff ( x in dom f & a < f . x ) );

existence

proof end;

uniqueness

proof end;

defpred S₄[ object ] means ( $1 in dom f & a <= f . $1 );

func great_eq_dom (f,a) -> set means :Def14: :: MESFUNC1:def 13
for x being object holds
( x in it iff ( x in dom f & a <= f . x ) );

existence

proof end;

uniqueness

proof end;

redefine func Coim (f,a) means :Def15: :: MESFUNC1:def 14
for x being set holds
( x in it iff ( x in dom f & f . x = a ) );

compatibility

proof end;

end;

:: deftheorem Def11 defines less_dom MESFUNC1:def 10 :

:: deftheorem Def12 defines less_eq_dom MESFUNC1:def 11 :

:: deftheorem Def13 defines great_dom MESFUNC1:def 12 :

:: deftheorem Def14 defines great_eq_dom MESFUNC1:def 13 :

:: deftheorem Def15 defines eq_dom MESFUNC1:def 14 :

definition

let X be set ;
let f be PartFunc of X,ExtREAL;
let a be ExtReal;
:: original: less_dom
redefine func less_dom (f,a) -> Subset of X;
coherence

proof end;

:: original: less_eq_dom
redefine func less_eq_dom (f,a) -> Subset of X;
coherence

proof end;

:: original: great_dom
redefine func great_dom (f,a) -> Subset of X;
coherence

proof end;

:: original: great_eq_dom
redefine func great_eq_dom (f,a) -> Subset of X;
coherence

proof end;

:: original: eq_dom
redefine func eq_dom (f,a) -> Subset of X;
coherence

proof end;

end;

theorem Th14: :: MESFUNC1:14

for X being set
for f being PartFunc of X,ExtREAL
for A being set
for a being ExtReal st A c= dom f holds
A /\ (great_eq_dom (f,a)) = A \ (A /\ (less_dom (f,a)))

proof end;

theorem Th15: :: MESFUNC1:15

for X being set
for f being PartFunc of X,ExtREAL
for A being set
for a being ExtReal st A c= dom f holds
A /\ (great_dom (f,a)) = A \ (A /\ (less_eq_dom (f,a)))

proof end;

theorem Th16: :: MESFUNC1:16

for X being set
for f being PartFunc of X,ExtREAL
for A being set
for a being ExtReal st A c= dom f holds
A /\ (less_eq_dom (f,a)) = A \ (A /\ (great_dom (f,a)))

proof end;

theorem Th17: :: MESFUNC1:17

for X being set
for f being PartFunc of X,ExtREAL
for A being set
for a being ExtReal st A c= dom f holds
A /\ (less_dom (f,a)) = A \ (A /\ (great_eq_dom (f,a)))

proof end;

theorem :: MESFUNC1:18

for X being set
for f being PartFunc of X,ExtREAL
for A being set
for a being ExtReal holds A /\ (eq_dom (f,a)) = (A /\ (great_eq_dom (f,a))) /\ (less_eq_dom (f,a))

proof end;

theorem :: MESFUNC1:19

for X being set
for S being SigmaField of X
for F being sequence of S
for f being PartFunc of X,ExtREAL
for A being set
for r being Real st ( for n being Element of NAT holds F . n = A /\ (great_dom (f,(r - (1 / (n + 1))))) ) holds
A /\ (great_eq_dom (f,r)) = meet (rng F)

proof end;

theorem Th20: :: MESFUNC1:20

for X being set
for f being PartFunc of X,ExtREAL
for S being SigmaField of X
for F being sequence of S
for A being set
for r being Real st ( for n being Element of NAT holds F . n = A /\ (less_dom (f,(r + (1 / (n + 1))))) ) holds
A /\ (less_eq_dom (f,r)) = meet (rng F)

proof end;

theorem Th21: :: MESFUNC1:21

for X being set
for f being PartFunc of X,ExtREAL
for S being SigmaField of X
for F being sequence of S
for A being set
for r being Real st ( for n being Element of NAT holds F . n = A /\ (less_eq_dom (f,(r - (1 / (n + 1))))) ) holds
A /\ (less_dom (f,r)) = union (rng F)

proof end;

theorem Th22: :: MESFUNC1:22

for X being set
for S being SigmaField of X
for F being sequence of S
for f being PartFunc of X,ExtREAL
for A being set
for r being Real st ( for n being Element of NAT holds F . n = A /\ (great_eq_dom (f,(r + (1 / (n + 1))))) ) holds
A /\ (great_dom (f,r)) = union (rng F)

proof end;

theorem Th23: :: MESFUNC1:23

for X being set
for S being SigmaField of X
for F being sequence of S
for f being PartFunc of X,ExtREAL
for A being set st ( for n being Element of NAT holds F . n = A /\ (great_dom (f,n)) ) holds
A /\ (eq_dom (f,+infty)) = meet (rng F)

proof end;

theorem Th24: :: MESFUNC1:24

for X being set
for S being SigmaField of X
for F being sequence of S
for f being PartFunc of X,ExtREAL
for A being set st ( for n being Element of NAT holds F . n = A /\ (less_dom (f,n)) ) holds
A /\ (less_dom (f,+infty)) = union (rng F)

proof end;

theorem Th25: :: MESFUNC1:25

for X being set
for S being SigmaField of X
for F being sequence of S
for f being PartFunc of X,ExtREAL
for A being set st ( for n being Nat holds F . n = A /\ (less_dom (f,(- n))) ) holds
A /\ (eq_dom (f,-infty)) = meet (rng F)

proof end;

theorem Th26: :: MESFUNC1:26

for X being set
for S being SigmaField of X
for F being sequence of S
for f being PartFunc of X,ExtREAL
for A being set st ( for n being Element of NAT holds F . n = A /\ (great_dom (f,(- n))) ) holds
A /\ (great_dom (f,-infty)) = union (rng F)

proof end;

definition

let X be non empty set ;
let S be SigmaField of X;
let A be Element of S;
let f be PartFunc of X,ExtREAL;

attr f is A -measurable means :: MESFUNC1:def 15
for r being Real holds A /\ (less_dom (f,r)) in S;

end;

:: deftheorem defines -measurable MESFUNC1:def 15 :

theorem :: MESFUNC1:27

for X being non empty set
for S being SigmaField of X
for f being PartFunc of X,ExtREAL
for A being Element of S st A c= dom f holds
( f is A -measurable iff for r being Real holds A /\ (great_eq_dom (f,r)) in S )

proof end;

theorem Th28: :: MESFUNC1:28

for X being non empty set
for S being SigmaField of X
for f being PartFunc of X,ExtREAL
for A being Element of S holds
( f is A -measurable iff for r being Real holds A /\ (less_eq_dom (f,r)) in S )

proof end;

theorem Th29: :: MESFUNC1:29

proof end;

theorem :: MESFUNC1:30

for X being non empty set
for S being SigmaField of X
for f being PartFunc of X,ExtREAL
for A, B being Element of S st B c= A & f is A -measurable holds
f is B -measurable

proof end;

theorem :: MESFUNC1:31

for X being non empty set
for S being SigmaField of X
for f being PartFunc of X,ExtREAL
for A, B being Element of S st f is A -measurable & f is B -measurable holds
f is A \/ B -measurable

proof end;

theorem :: MESFUNC1:32

for X being non empty set
for S being SigmaField of X
for f being PartFunc of X,ExtREAL
for A being Element of S
for r, s being Real st f is A -measurable & A c= dom f holds
(A /\ (great_dom (f,r))) /\ (less_dom (f,s)) in S

proof end;

theorem :: MESFUNC1:33

for X being non empty set
for S being SigmaField of X
for f being PartFunc of X,ExtREAL
for A being Element of S st f is A -measurable & A c= dom f holds
A /\ (eq_dom (f,+infty)) in S

proof end;

theorem :: MESFUNC1:34

for X being non empty set
for S being SigmaField of X
for f being PartFunc of X,ExtREAL
for A being Element of S st f is A -measurable holds
A /\ (eq_dom (f,-infty)) in S

proof end;

theorem :: MESFUNC1:35

for X being non empty set
for S being SigmaField of X
for f being PartFunc of X,ExtREAL
for A being Element of S st f is A -measurable & A c= dom f holds
(A /\ (great_dom (f,-infty))) /\ (less_dom (f,+infty)) in S

proof end;

theorem :: MESFUNC1:36

for X being non empty set
for S being SigmaField of X
for f, g being PartFunc of X,ExtREAL
for A being Element of S
for r being Real st f is A -measurable & g is A -measurable & A c= dom g holds
(A /\ (less_dom (f,r))) /\ (great_dom (g,r)) in S

proof end;

theorem :: MESFUNC1:37

for X being non empty set
for S being SigmaField of X
for f being PartFunc of X,ExtREAL
for A being Element of S
for r being Real st f is A -measurable & A c= dom f holds
r (#) f is A -measurable

proof end;