let X be non empty set ; :: thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL st ex A being Element of S st
( A = dom f & f is_measurable_on A ) & f is nonnegative holds
0 <= Integral M,f
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f being PartFunc of X,REAL st ex A being Element of S st
( A = dom f & f is_measurable_on A ) & f is nonnegative holds
0 <= Integral M,f
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,REAL st ex A being Element of S st
( A = dom f & f is_measurable_on A ) & f is nonnegative holds
0 <= Integral M,f
let f be PartFunc of X,REAL ; :: thesis: ( ex A being Element of S st
( A = dom f & f is_measurable_on A ) & f is nonnegative implies 0 <= Integral M,f )
assume A1: ( ex A being Element of S st
( A = dom f & f is_measurable_on A ) & f is nonnegative ) ; :: thesis: 0 <= Integral M,f
then consider A being Element of S such that
A2: ( A = dom f & f is_measurable_on A ) ;
( A = dom (R_EAL f) & R_EAL f is_measurable_on A ) by A2, Def6;
hence 0 <= Integral M,f by A1, MESFUNC5:96; :: thesis: verum