let X be non empty set ; :: thesis: for S being SigmaField of X
for M being sigma_Measure of S
for F being SetSequence of S st M . (Union F) < +infty holds
ex G being sequence of S st
( G = superior_setsequence F & M . (lim_sup F) = inf (rng (M * G)) )
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for F being SetSequence of S st M . (Union F) < +infty holds
ex G being sequence of S st
( G = superior_setsequence F & M . (lim_sup F) = inf (rng (M * G)) )
let M be sigma_Measure of S; :: thesis: for F being SetSequence of S st M . (Union F) < +infty holds
ex G being sequence of S st
( G = superior_setsequence F & M . (lim_sup F) = inf (rng (M * G)) )
let F be SetSequence of S; :: thesis: ( M . (Union F) < +infty implies ex G being sequence of S st
( G = superior_setsequence F & M . (lim_sup F) = inf (rng (M * G)) ) )
rng (superior_setsequence F) c= S ;
then reconsider G = superior_setsequence F as sequence of S by FUNCT_2:6;
reconsider F1 = F, G1 = G as SetSequence of X ;
A1: for n being Nat holds G . (n + 1) c= G . n by NAT_1:12, PROB_1:def 4;
G . 0 = union { (F . k) where k is Nat : 0 <= k } by SETLIM_1:def 3;
then A2: G . 0 = union (rng F) by SETLIM_1:5;
consider f being SetSequence of X such that
A3: lim_sup F1 = meet f and
A4: for n being Nat holds f . n = Union (F1 ^\ n) by KURATO_0:def 2;

now :: thesis: for n being Element of NAT holds f . n = G1 . n

let n be Element of NAT ; :: thesis: f . n = G1 . n
f . n = Union (F1 ^\ n) by A4;
hence f . n = G1 . n by Th2; :: thesis: verum

end;

then A5: f = G1 by FUNCT_2:63;
assume M . (Union F) < +infty ; :: thesis: ex G being sequence of S st
( G = superior_setsequence F & M . (lim_sup F) = inf (rng (M * G)) )
then M . (meet (rng G)) = inf (rng (M * G)) by A1, A2, MEASURE3:12;
hence ex G being sequence of S st
( G = superior_setsequence F & M . (lim_sup F) = inf (rng (M * G)) ) by A3, A5; :: thesis: verum