let X be non empty set ; :: thesis: for f being PartFunc of X,ExtREAL
for S being SigmaField of X
for A being Element of S st f is_measurable_on A & A c= dom f holds
max- f is_measurable_on A
let f be PartFunc of X,ExtREAL ; :: thesis: for S being SigmaField of X
for A being Element of S st f is_measurable_on A & A c= dom f holds
max- f is_measurable_on A
let S be SigmaField of X; :: thesis: for A being Element of S st f is_measurable_on A & A c= dom f holds
max- f is_measurable_on A
let A be Element of S; :: thesis: ( f is_measurable_on A & A c= dom f implies max- f is_measurable_on A )
assume that
A1: f is_measurable_on A and
A2: A c= dom f ; :: thesis: max- f is_measurable_on A
for r being real number holds A /\ (less_dom (max- f),(R_EAL r)) in S

proof

let r be real number ; :: thesis: A /\ (less_dom (max- f),(R_EAL r)) in S
reconsider r = r as Real by XREAL_0:def 1;

now

per cases ( 0 < r or r <= 0 ) ;

suppose A3: 0 < r ; :: thesis: A /\ (less_dom (max- f),(R_EAL r)) in S

(- 1) (#) f is_measurable_on A by A1, A2, MESFUNC1:41;
then A4: - f is_measurable_on A by Th11;
less_dom (max- f),(R_EAL r) = less_dom (- f),(R_EAL r)

proof

for x being set st x in less_dom (max- f),(R_EAL r) holds
x in less_dom (- f),(R_EAL r)

proof

let x be set ; :: thesis: ( x in less_dom (max- f),(R_EAL r) implies x in less_dom (- f),(R_EAL r) )
assume A5: x in less_dom (max- f),(R_EAL r) ; :: thesis: x in less_dom (- f),(R_EAL r)
then A6: ( x in dom (max- f) & (max- f) . x < R_EAL r ) by MESFUNC1:def 12;
reconsider x = x as Element of X by A5;
A7: max (- (f . x)),0. < R_EAL r by A6, Def3;
then A8: ( - (f . x) <= R_EAL r & 0. <= R_EAL r ) by XXREAL_0:30;
- (f . x) <> R_EAL r

proof

assume A9: - (f . x) = R_EAL r ; :: thesis: contradiction
then max (- (f . x)),0. = 0. by A7, XXREAL_0:16;
hence contradiction by A7, A9, XXREAL_0:def 10; :: thesis: verum

end;

then A10: - (f . x) < R_EAL r by A8, XXREAL_0:1;
x in dom f by A6, Def3;
then A11: x in dom (- f) by MESFUNC1:def 7;
then (- f) . x = - (f . x) by MESFUNC1:def 7;
hence x in less_dom (- f),(R_EAL r) by A10, A11, MESFUNC1:def 12; :: thesis: verum

end;

then A12: less_dom (max- f),(R_EAL r) c= less_dom (- f),(R_EAL r) by TARSKI:def 3;
for x being set st x in less_dom (- f),(R_EAL r) holds
x in less_dom (max- f),(R_EAL r)

proof

let x be set ; :: thesis: ( x in less_dom (- f),(R_EAL r) implies x in less_dom (max- f),(R_EAL r) )
assume A13: x in less_dom (- f),(R_EAL r) ; :: thesis: x in less_dom (max- f),(R_EAL r)
then A14: ( x in dom (- f) & (- f) . x < R_EAL r ) by MESFUNC1:def 12;
A15: (- f) . x < R_EAL r by A13, MESFUNC1:def 12;
reconsider x = x as Element of X by A13;
x in dom f by A14, MESFUNC1:def 7;
then A16: x in dom (max- f) by Def3;

now

per cases ( 0. <= - (f . x) or not 0. <= - (f . x) ) ;

suppose 0. <= - (f . x) ; :: thesis: x in less_dom (max- f),(R_EAL r)

then max (- (f . x)),0. = - (f . x) by XXREAL_0:def 10;
then (max- f) . x = - (f . x) by A16, Def3
.= (- f) . x by A14, MESFUNC1:def 7 ;
hence x in less_dom (max- f),(R_EAL r) by A15, A16, MESFUNC1:def 12; :: thesis: verum

end;

suppose not 0. <= - (f . x) ; :: thesis: x in less_dom (max- f),(R_EAL r)

then max (- (f . x)),0. = 0. by XXREAL_0:def 10;
then (max- f) . x = 0. by A16, Def3;
hence x in less_dom (max- f),(R_EAL r) by A3, A16, MESFUNC1:def 12; :: thesis: verum

end;

end;

end;

hence x in less_dom (max- f),(R_EAL r) ; :: thesis: verum

end;

then less_dom (- f),(R_EAL r) c= less_dom (max- f),(R_EAL r) by TARSKI:def 3;
hence less_dom (max- f),(R_EAL r) = less_dom (- f),(R_EAL r) by A12, XBOOLE_0:def 10; :: thesis: verum

end;

hence A /\ (less_dom (max- f),(R_EAL r)) in S by A4, MESFUNC1:def 17; :: thesis: verum

end;

suppose A17: r <= 0 ; :: thesis: A /\ (less_dom (max- f),(R_EAL r)) in S

for x being Element of X holds not x in less_dom (max- f),(R_EAL r)

proof

let x be Element of X; :: thesis: not x in less_dom (max- f),(R_EAL r)
assume x in less_dom (max- f),(R_EAL r) ; :: thesis: contradiction
then A18: ( x in dom (max- f) & (max- f) . x < R_EAL r ) by MESFUNC1:def 12;
then (max- f) . x = max (- (f . x)),0. by Def3;
hence contradiction by A17, A18, XXREAL_0:25; :: thesis: verum

end;

then less_dom (max- f),(R_EAL r) = {} by SUBSET_1:10;
hence A /\ (less_dom (max- f),(R_EAL r)) in S by PROB_1:10; :: thesis: verum

end;

end;

end;

hence A /\ (less_dom (max- f),(R_EAL r)) in S ; :: thesis: verum

end;

hence max- f is_measurable_on A by MESFUNC1:def 17; :: thesis: verum