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 /\ (eq_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 /\ (eq_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 /\ (eq_dom (f,+infty)) in S
let A be Element of S; :: thesis: ( f is_measurable_on A & A c= dom f implies A /\ (eq_dom (f,+infty)) in S )
assume A1: ( f is_measurable_on A & A c= dom f ) ; :: thesis: A /\ (eq_dom (f,+infty)) in S
defpred S₁[ Element of NAT , set ] means A /\ (great_dom (f,(R_EAL $1))) = $2;
A2: 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, Th33;
take y ; :: thesis: S₁[n,y]
thus S₁[n,y] ; :: thesis: verum

end;

consider F being Function of NAT,S such that
A3: for n being Element of NAT holds S₁[n,F . n] from FUNCT_2:sch 3(A2);
A /\ (eq_dom (f,+infty)) = meet (rng F) by A3, Th27;
hence A /\ (eq_dom (f,+infty)) in S ; :: thesis: verum