reserve a,b,r for Real;
reserve A,B for non empty set;
reserve f,g,h for Element of PFuncs(A,REAL);
reserve u,v,w for VECTOR of RLSp_PFunctA;
reserve X for non empty set,
  x for Element of X,
  S for SigmaField of X,
  M for sigma_Measure of S,
  E,E1,E2 for Element of S,
  f,g,h,f1,g1 for PartFunc of X ,REAL;
reserve v,u for VECTOR of RLSp_L1Funct M;
reserve v,u for VECTOR of RLSp_AlmostZeroFunct M;
reserve x for Point of Pre-L-Space M;
reserve x,y for Point of L-1-Space M;

theorem Th52:
  E = dom f & (for x be set st x in dom f holds f.x=r) implies f
  is E-measurable
proof
  assume
A1: E = dom f;
  r in REAL by XREAL_0:def 1;
  then reconsider r0=r as R_eal by NUMBERS:31;
  set g=R_EAL f;
  consider g0 be PartFunc of X,ExtREAL such that
A2: g0 is_simple_func_in S and
A3: dom g0 = E and
A4: for x be object st x in E holds g0.x=r0 by MESFUNC5:41;
  assume
A5: for x be set st x in dom f holds f.x=r;
  now
    let x be Element of X;
    assume
A6: x in dom g;
    then g.x = r by A5;
    hence g.x = g0.x by A1,A4,A6;
  end;
  then g0=g by A1,A3,PARTFUN1:5;
  then g is E-measurable by A2,MESFUNC2:34;
  hence thesis;
end;
