let X be non empty set ; :: thesis: for S being SigmaField of X
for f being PartFunc of X,ExtREAL
for A being Element of S st f is_measurable_on A & A c= dom f holds
(A /\ (great_dom f,-infty )) /\ (less_dom f,+infty ) in S
let S be SigmaField of X; :: thesis: for f being PartFunc of X,ExtREAL
for A being Element of S st f is_measurable_on A & A c= dom f holds
(A /\ (great_dom f,-infty )) /\ (less_dom f,+infty ) in S
let f be PartFunc of X,ExtREAL ; :: thesis: for A being Element of S st f is_measurable_on A & A c= dom f holds
(A /\ (great_dom f,-infty )) /\ (less_dom f,+infty ) in S
let A be Element of S; :: thesis: ( f is_measurable_on A & A c= dom f implies (A /\ (great_dom f,-infty )) /\ (less_dom f,+infty ) in S )
assume that
A1: f is_measurable_on A and
A2: A c= dom f ; :: thesis: (A /\ (great_dom f,-infty )) /\ (less_dom f,+infty ) in S
A3: A /\ (great_dom f,-infty ) in S

proof

defpred S₁[ Element of NAT , set ] means A /\ (great_dom f,(R_EAL (- $1))) = $2;
A4: for n being Element of NAT ex y being Element of S st S₁[n,y]

proof

let n be Element of NAT ; :: thesis: ex y being Element of S st S₁[n,y]
reconsider y = A /\ (great_dom f,(R_EAL (- n))) as Element of S by A1, A2, Th33;
take y ; :: thesis: S₁[n,y]
thus S₁[n,y] ; :: thesis: verum

end;

consider F being Function of NAT ,S such that
A5: for n being Element of NAT holds S₁[n,F . n] from FUNCT_2:sch 3(A4);
A6: A /\ (great_dom f,-infty ) = union (rng F) by A5, Th30;
thus A /\ (great_dom f,-infty ) in S by A6; :: thesis: verum

end;

A7: A /\ (less_dom f,+infty ) in S

proof

defpred S₁[ Element of NAT , set ] means A /\ (less_dom f,(R_EAL $1)) = $2;
A8: for n being Element of NAT ex y being Element of S st S₁[n,y]

proof

let n be Element of NAT ; :: thesis: ex y being Element of S st S₁[n,y]
reconsider y = A /\ (less_dom f,(R_EAL n)) as Element of S by A1, Def17;
take y ; :: thesis: S₁[n,y]
thus S₁[n,y] ; :: thesis: verum

end;

consider F being Function of NAT ,S such that
A9: for n being Element of NAT holds S₁[n,F . n] from FUNCT_2:sch 3(A8);
A10: A /\ (less_dom f,+infty ) = union (rng F) by A9, Th28;
thus A /\ (less_dom f,+infty ) in S by A10; :: thesis: verum

end;

A11: (A /\ (great_dom f,-infty )) /\ (A /\ (less_dom f,+infty )) = ((A /\ (great_dom f,-infty )) /\ A) /\ (less_dom f,+infty ) by XBOOLE_1:16
.= ((great_dom f,-infty ) /\ (A /\ A)) /\ (less_dom f,+infty ) by XBOOLE_1:16 ;
thus (A /\ (great_dom f,-infty )) /\ (less_dom f,+infty ) in S by A3, A7, A11, FINSUB_1:def 2; :: thesis: verum