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 A -measurable ) holds
( f is_integrable_on M iff abs f is_integrable_on M )
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 A -measurable ) holds
( f is_integrable_on M iff abs f is_integrable_on M )
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 A -measurable ) holds
( f is_integrable_on M iff abs f is_integrable_on M )
let f be PartFunc of X,REAL; :: thesis: ( ex A being Element of S st
( A = dom f & f is A -measurable ) implies ( f is_integrable_on M iff abs f is_integrable_on M ) )
assume ex A being Element of S st
( A = dom f & f is A -measurable ) ; :: thesis: ( f is_integrable_on M iff abs f is_integrable_on M )
then consider A being Element of S such that
A1: A = dom f and
A2: f is A -measurable ;
R_EAL f is A -measurable by A2;
then ( R_EAL f is_integrable_on M iff |.(R_EAL f).| is_integrable_on M ) by A1, MESFUNC5:100;
then ( f is_integrable_on M iff R_EAL (abs f) is_integrable_on M ) by Th1;
hence ( f is_integrable_on M iff abs f is_integrable_on M ) ; :: thesis: verum