reserve X for non empty set,
        x for Element of X,
        S for SigmaField of X,
        M for sigma_Measure of S,
        f,g,f1,g1 for PartFunc of X,REAL,
        l,m,n,n1,n2 for Nat,
        a,b,c for Real;
reserve k for positive Real;

theorem Th23:
X --> 0 is PartFunc of X,REAL & X --> 0 in Lp_Functions(M,k)
proof
   reconsider g = X --> In(0,REAL) as Function of X,REAL by FUNCOP_1:46;
   reconsider ND = X as Element of S by MEASURE1:34;
   ND` = {} by XBOOLE_1:37; then
A1:M.ND` = 0 by VALUED_0:def 19;
A2:dom g = X by FUNCT_2:def 1;
   for x be Element of X st x in dom g holds g.x = 0 by FUNCOP_1:7; then
A3:g is_integrable_on M by A2,Th15;
   for x be object st x in dom g holds 0 <= g.x; then
   abs g = g by Th14,MESFUNC6:52; then
A4:(abs g) to_power k = g by Th12;
   for x be set st x in dom g holds g.x = 0 by FUNCOP_1:7; then
   g is ND-measurable by A2,LPSPACE1:52;
   hence thesis by A1,A2,A3,A4;
end;
