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 ex G being sequence of 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 S ex G being sequence of 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 S ex G being sequence of S st
( G = inferior_setsequence F & M . (lim_inf F) = sup (rng (M * G)) )
let F be SetSequence of S; :: thesis: ex G being sequence of S st
( G = inferior_setsequence F & M . (lim_inf F) = sup (rng (M * G)) )
rng (inferior_setsequence F) c= S ;
then reconsider G = inferior_setsequence F as sequence of S by FUNCT_2:6;
for n being Nat holds G . n c= G . (n + 1) by PROB_1:def 5, NAT_1:12;
then M . (union (rng G)) = sup (rng (M * G)) by MEASURE2:23;
hence ex G being sequence of S st
( G = inferior_setsequence F & M . (lim_inf F) = sup (rng (M * G)) ) ; :: thesis: verum