:: MEASURE4 semantic presentation

theorem Th1: :: MEASURE4:1
for x, y, z being R_eal st 0. <= x & 0. <= y & 0. <= z holds
(x + y) + z = x + (y + z)
proof end;

theorem Th2: :: MEASURE4:2
for x, y, z being R_eal st not x = -infty & not x = +infty holds
( y + x <= z iff y <= z - x )
proof end;

theorem Th3: :: MEASURE4:3
canceled;

theorem Th4: :: MEASURE4:4
for X being set
for S being non empty Subset-Family of X
for F, G being Function of NAT ,S
for A being Element of S st ( for n being Element of NAT holds G . n = A /\ (F . n) ) holds
union (rng G) = A /\ (union (rng F))
proof end;

theorem Th5: :: MEASURE4:5
for X being set
for S being non empty Subset-Family of X
for F, G being Function of NAT ,S st G . 0 = F . 0 & ( for n being Element of NAT holds G . (n + 1) = (F . (n + 1)) \/ (G . n) ) holds
for H being Function of NAT ,S st H . 0 = F . 0 & ( for n being Element of NAT holds H . (n + 1) = (F . (n + 1)) \ (G . n) ) holds
union (rng F) = union (rng H)
proof end;

theorem Th6: :: MEASURE4:6
for X being set holds bool X is SigmaField of X
proof end;

theorem Th7: :: MEASURE4:7
for X being set
for a, b being R_eal ex M being Function of bool X, ExtREAL st
for A being Subset of X holds
( ( A = {} implies M . A = a ) & ( A <> {} implies M . A = b ) )
proof end;

theorem Th8: :: MEASURE4:8
for X being set ex M being Function of bool X, ExtREAL st
for A being Subset of X holds M . A = 0.
proof end;

theorem Th9: :: MEASURE4:9
canceled;

theorem Th10: :: MEASURE4:10
canceled;

theorem Th11: :: MEASURE4:11
for X being set ex M being Function of bool X, ExtREAL st
( M is nonnegative & M . {} = 0. & ( for A, B being Subset of X st A c= B holds
( M . A <= M . B & ( for F being Function of NAT , bool X holds M . (union (rng F)) <= SUM (M * F) ) ) ) )
proof end;

definition
let X be set ;
canceled;
mode C_Measure of c1 -> Function of bool a1, ExtREAL means :Def2: :: MEASURE4:def 2
( it is nonnegative & it . {} = 0. & ( for A, B being Subset of X st A c= B holds
( it . A <= it . B & ( for F being Function of NAT , bool X holds it . (union (rng F)) <= SUM (it * F) ) ) ) );
existence
ex b1 being Function of bool X, ExtREAL st
( b1 is nonnegative & b1 . {} = 0. & ( for A, B being Subset of X st A c= B holds
( b1 . A <= b1 . B & ( for F being Function of NAT , bool X holds b1 . (union (rng F)) <= SUM (b1 * F) ) ) ) )
by ;
end;

:: deftheorem Def1 MEASURE4:def 1 :
canceled;

:: deftheorem Def2 defines C_Measure MEASURE4:def 2 :
for X being set
for b2 being Function of bool X, ExtREAL holds
( b2 is C_Measure of X iff ( b2 is nonnegative & b2 . {} = 0. & ( for A, B being Subset of X st A c= B holds
( b2 . A <= b2 . B & ( for F being Function of NAT , bool X holds b2 . (union (rng F)) <= SUM (b2 * F) ) ) ) ) );

definition
let X be set ;
let C be C_Measure of X;
func sigma_Field c2 -> non empty Subset-Family of a1 means :Def3: :: MEASURE4:def 3
for A being Subset of X holds
( A in it iff for W, Z being Subset of X st W c= A & Z c= X \ A holds
(C . W) + (C . Z) <= C . (W \/ Z) );
existence
ex b1 being non empty Subset-Family of X st
for A being Subset of X holds
( A in b1 iff for W, Z being Subset of X st W c= A & Z c= X \ A holds
(C . W) + (C . Z) <= C . (W \/ Z) )
proof end;
uniqueness
for b1, b2 being non empty Subset-Family of X st ( for A being Subset of X holds
( A in b1 iff for W, Z being Subset of X st W c= A & Z c= X \ A holds
(C . W) + (C . Z) <= C . (W \/ Z) ) ) & ( for A being Subset of X holds
( A in b2 iff for W, Z being Subset of X st W c= A & Z c= X \ A holds
(C . W) + (C . Z) <= C . (W \/ Z) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def3 defines sigma_Field MEASURE4:def 3 :
for X being set
for C being C_Measure of X
for b3 being non empty Subset-Family of X holds
( b3 = sigma_Field C iff for A being Subset of X holds
( A in b3 iff for W, Z being Subset of X st W c= A & Z c= X \ A holds
(C . W) + (C . Z) <= C . (W \/ Z) ) );

theorem Th12: :: MEASURE4:12
for X being set
for C being C_Measure of X
for W, Z being Subset of X holds C . (W \/ Z) <= (C . W) + (C . Z)
proof end;

theorem Th13: :: MEASURE4:13
canceled;

theorem Th14: :: MEASURE4:14
for X being set
for C being C_Measure of X
for A being Subset of X holds
( A in sigma_Field C iff for W, Z being Subset of X st W c= A & Z c= X \ A holds
(C . W) + (C . Z) = C . (W \/ Z) )
proof end;

theorem Th15: :: MEASURE4:15
for X being set
for C being C_Measure of X
for W, Z being Subset of X st W in sigma_Field C & Z in sigma_Field C & Z misses W holds
C . (W \/ Z) = (C . W) + (C . Z)
proof end;

theorem Th16: :: MEASURE4:16
for A, X being set
for C being C_Measure of X st A in sigma_Field C holds
X \ A in sigma_Field C
proof end;

theorem Th17: :: MEASURE4:17
for X, A, B being set
for C being C_Measure of X st A in sigma_Field C & B in sigma_Field C holds
A \/ B in sigma_Field C
proof end;

theorem Th18: :: MEASURE4:18
for X, A, B being set
for C being C_Measure of X st A in sigma_Field C & B in sigma_Field C holds
A /\ B in sigma_Field C
proof end;

theorem Th19: :: MEASURE4:19
for X, A, B being set
for C being C_Measure of X st A in sigma_Field C & B in sigma_Field C holds
A \ B in sigma_Field C
proof end;

theorem Th20: :: MEASURE4:20
for X being set
for S being SigmaField of X
for N being Function of NAT ,S
for A being Element of S ex F being Function of NAT ,S st
for n being Element of NAT holds F . n = A /\ (N . n)
proof end;

theorem Th21: :: MEASURE4:21
for X being set
for C being C_Measure of X holds sigma_Field C is SigmaField of X
proof end;

registration
let X be set ;
let C be C_Measure of X;
cluster sigma_Field a2 -> non empty compl-closed sigma-additive ;
coherence
( sigma_Field C is sigma-additive & sigma_Field C is compl-closed & not sigma_Field C is empty )
by ;
end;

definition
let X be set ;
let S be SigmaField of X;
let A be N_Sub_fam of S;
redefine func union as union c3 -> Element of a2;
coherence
union A is Element of S
proof end;
end;

definition
let X be set ;
let C be C_Measure of X;
func sigma_Meas c2 -> Function of sigma_Field a2, ExtREAL means :Def4: :: MEASURE4:def 4
for A being Subset of X st A in sigma_Field C holds
it . A = C . A;
existence
ex b1 being Function of sigma_Field C, ExtREAL st
for A being Subset of X st A in sigma_Field C holds
b1 . A = C . A
proof end;
uniqueness
for b1, b2 being Function of sigma_Field C, ExtREAL st ( for A being Subset of X st A in sigma_Field C holds
b1 . A = C . A ) & ( for A being Subset of X st A in sigma_Field C holds
b2 . A = C . A ) holds
b1 = b2
proof end;
end;

:: deftheorem Def4 defines sigma_Meas MEASURE4:def 4 :
for X being set
for C being C_Measure of X
for b3 being Function of sigma_Field C, ExtREAL holds
( b3 = sigma_Meas C iff for A being Subset of X st A in sigma_Field C holds
b3 . A = C . A );

theorem Th22: :: MEASURE4:22
for X being set
for C being C_Measure of X holds sigma_Meas C is Measure of sigma_Field C
proof end;

definition
let X be set ;
let C be C_Measure of X;
let A be Element of sigma_Field C;
redefine func . as c2 . c3 -> R_eal;
coherence
C . A is R_eal
by FUNCT_2:7;
end;

theorem Th23: :: MEASURE4:23
for X being set
for C being C_Measure of X holds sigma_Meas C is sigma_Measure of sigma_Field C
proof end;

definition
let X be set ;
let C be C_Measure of X;
redefine func sigma_Meas as sigma_Meas c2 -> sigma_Measure of sigma_Field a2;
coherence
sigma_Meas C is sigma_Measure of sigma_Field C
by Th1;
end;

theorem Th24: :: MEASURE4:24
for X being set
for C being C_Measure of X
for A being Subset of X st C . A = 0. holds
A in sigma_Field C
proof end;

theorem Th25: :: MEASURE4:25
for X being set
for C being C_Measure of X holds sigma_Meas C is_complete sigma_Field C
proof end;