
theorem Th1:
for X being non empty set, S being SigmaField of X, M being sigma_Measure of S,
 f being PartFunc of X,ExtREAL st dom f = {} holds Integral(M,f) = 0
proof
    let X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
    f be PartFunc of X,ExtREAL;
    assume
A1:  dom f = {}; then
    reconsider E = dom f as Element of S by MEASURE1:7;
A2: E = dom(max+f) & E = dom(max-f) by MESFUNC2:def 2,def 3;
A3: M.E = 0 by A1,VALUED_0:def 19;
    max+f is E-measurable & max-f is E-measurable by A1; then
A4: integral+(M,(max+f)|E) = 0 & integral+(M,(max-f)|E) = 0
     by A2,A3,MESFUNC5:82,MESFUN11:5; then
    Integral(M,f) = integral+(M,max+f)-0 by A2,MESFUNC5:def 16;
    hence thesis by A2,A4,XXREAL_3:15;
end;
