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 ex G being Function of NAT ,S st
( G = inferior_setsequence F & M . (lim_inf F) = sup (rng (M * G)) )
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for F being SetSequence of ex G being Function of NAT ,S st
( G = inferior_setsequence F & M . (lim_inf F) = sup (rng (M * G)) )
let M be sigma_Measure of S; :: thesis: for F being SetSequence of ex G being Function of NAT ,S st
( G = inferior_setsequence F & M . (lim_inf F) = sup (rng (M * G)) )
let F be SetSequence of ; :: thesis: ex G being Function of NAT ,S st
( G = inferior_setsequence F & M . (lim_inf F) = sup (rng (M * G)) )
rng (inferior_setsequence F) c= S by RELAT_1:def 19;
then reconsider G = inferior_setsequence F as Function of NAT ,S by FUNCT_2:8;

now

let n be Element of NAT ; :: thesis: G . n c= G . (n + 1)
n <= n + 1 by NAT_1:12;
hence G . n c= G . (n + 1) by PROB_1:def 7; :: thesis: verum

end;

then M . (union (rng G)) = sup (rng (M * G)) by MEASURE2:27;
hence ex G being Function of NAT ,S st
( G = inferior_setsequence F & M . (lim_inf F) = sup (rng (M * G)) ) ; :: thesis: verum