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 f is_simple_func_in S & f is nonnegative holds
( Integral M,f = integral+ M,(R_EAL f) & Integral M,f = integral' M,(R_EAL f) )
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f being PartFunc of X,REAL st f is_simple_func_in S & f is nonnegative holds
( Integral M,f = integral+ M,(R_EAL f) & Integral M,f = integral' M,(R_EAL f) )
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,REAL st f is_simple_func_in S & f is nonnegative holds
( Integral M,f = integral+ M,(R_EAL f) & Integral M,f = integral' M,(R_EAL f) )
let f be PartFunc of X,REAL ; :: thesis: ( f is_simple_func_in S & f is nonnegative implies ( Integral M,f = integral+ M,(R_EAL f) & Integral M,f = integral' M,(R_EAL f) ) )
assume A1: ( f is_simple_func_in S & f is nonnegative ) ; :: thesis: ( Integral M,f = integral+ M,(R_EAL f) & Integral M,f = integral' M,(R_EAL f) )
then A2: ( R_EAL f is_simple_func_in S & R_EAL f is nonnegative ) by Th49;
then reconsider A = dom (R_EAL f) as Element of S by MESFUNC5:43;
R_EAL f is_measurable_on A by A2, MESFUNC2:37;
then ( f is_measurable_on A & A = dom f ) by Def6;
hence Integral M,f = integral+ M,(R_EAL f) by A1, Th82; :: thesis: Integral M,f = integral' M,(R_EAL f)
hence Integral M,f = integral' M,(R_EAL f) by A2, MESFUNC5:83; :: thesis: verum