Posted by oo By Lemma lim 0 ~(S\E) < E. We have to show that if' (xn ) ~"= ~ 0 - ~'.
5 applied to ~', lim if' (x n ) 2 if' (x o)' It thus remains to show that if' (xn ) -< if' Put By Lusin's theorem, is continuous on E. Hence n. 52 1f(x ) - 1f" (x ) < 1f(x ) - 1f" (x o ) + 25, n n 0 or 1f' (x ) < 1f' (x ) + 25 n for large 0 n. IIiii 1f' (x ) < 1f' (x ). 2 is complete. 1 we need the following characterization of continuity for potentials. 3: KC:rrr>. 4) ! x-y! <5 Ix-yl for every Proof: of compact support and given a compact set x € K. 4) for each As earlier, set K. J Fix € and choose 5 (t) ~, \ J, t>~ - J o