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,ExtREAL st f is_simple_func_in S & f is nonnegative holds
( Integral M,f = integral+ M,f & Integral M,f = integral' M,f )
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f being PartFunc of X,ExtREAL st f is_simple_func_in S & f is nonnegative holds
( Integral M,f = integral+ M,f & Integral M,f = integral' M,f )
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,ExtREAL st f is_simple_func_in S & f is nonnegative holds
( Integral M,f = integral+ M,f & Integral M,f = integral' M,f )
let f be PartFunc of X,ExtREAL ; :: thesis: ( f is_simple_func_in S & f is nonnegative implies ( Integral M,f = integral+ M,f & Integral M,f = integral' M,f ) )
assume that
A1: f is_simple_func_in S and
A2: f is nonnegative ; :: thesis: ( Integral M,f = integral+ M,f & Integral M,f = integral' M,f )
reconsider A = dom f as Element of S by A1, Th43;
f is_measurable_on A by A1, MESFUNC2:37;
hence Integral M,f = integral+ M,f by A2, Th94; :: thesis: Integral M,f = integral' M,f
hence Integral M,f = integral' M,f by A1, A2, Th83; :: thesis: verum