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 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 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 holds
A /\ (eq_dom (f,-infty)) in S
let A be Element of S; :: thesis: ( f is_measurable_on A implies A /\ (eq_dom (f,-infty)) in S )
assume A1: f is_measurable_on A ; :: thesis: A /\ (eq_dom (f,-infty)) in S
defpred S₁[ Element of NAT , set ] means A /\ (less_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 /\ (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
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, Th29;
hence A /\ (eq_dom (f,-infty)) in S ; :: thesis: verum