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 A -measurable 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 A -measurable 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 A -measurable holds
A /\ (eq_dom (f,-infty)) in S
let A be Element of S; :: thesis: ( f is A -measurable implies A /\ (eq_dom (f,-infty)) in S )
assume A1: f is A -measurable ; :: thesis: A /\ (eq_dom (f,-infty)) in S
defpred S₁[ Nat, set ] means A /\ (less_dom (f,(- $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,(- n))) as Element of S by A1;
take y ; :: thesis: S₁[n,y]
thus S₁[n,y] ; :: thesis: verum

end;

consider F being sequence of S such that
A3: for n being Element of NAT holds S₁[n,F . n] from FUNCT_2:sch 3(A2);
for n being Nat holds S₁[n,F . n]

proof

let n be Nat; :: thesis: S₁[n,F . n]
n in NAT by ORDINAL1:def 12;
hence S₁[n,F . n] by A3; :: thesis: verum

end;

then A /\ (eq_dom (f,-infty)) = meet (rng F) by Th25;
hence A /\ (eq_dom (f,-infty)) in S ; :: thesis: verum