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;
reserve v,u for VECTOR of RLSp_LpFunct(M,k);
reserve v,u for VECTOR of RLSp_AlmostZeroLpFunct(M,k);

theorem Th49:
f in Lp_Functions(M,k) implies
  Integral(M,(abs f) to_power k) in REAL & 0 <= Integral(M,(abs f) to_power k)
proof
   assume f in Lp_Functions(M,k); then
A1:ex f1 be PartFunc of X,REAL st
    f=f1 & ex ND be Element of S st M.ND` =0 & dom f1 = ND &
    f1 is ND-measurable & (abs f1) to_power k is_integrable_on M; then
   -infty < Integral(M,(abs f) to_power k) &
   Integral(M,(abs f) to_power k) < +infty by MESFUNC6:90;
   hence Integral(M,(abs f) to_power k) in REAL by XXREAL_0:14;
   R_EAL((abs f) to_power k) is_integrable_on M by A1; then
   consider A be Element of S such that
A2: A = dom(R_EAL((abs f) to_power k)) &
    R_EAL((abs f) to_power k) is A-measurable;
    A = dom((abs f) to_power k) &
   (abs f) to_power k is A-measurable by A2;
   hence thesis by MESFUNC6:84;
end;
