let a be Real; :: thesis: for X being non empty set
for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL st f in L1_Functions M holds
a (#) f in L1_Functions M
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 in L1_Functions M holds
a (#) f in L1_Functions M
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f being PartFunc of X,REAL st f in L1_Functions M holds
a (#) f in L1_Functions M
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,REAL st f in L1_Functions M holds
a (#) f in L1_Functions M
let f be PartFunc of X,REAL; :: thesis: ( f in L1_Functions M implies a (#) f in L1_Functions M )
set W = L1_Functions M;
assume f in L1_Functions M ; :: thesis: a (#) f in L1_Functions M
then ex f1 being PartFunc of X,REAL st
( f1 = f & ex ND being Element of S st
( M . ND = 0 & dom f1 = ND ` & f1 is_integrable_on M ) ) ;
then consider ND being Element of S such that
A1: M . ND = 0 and
A2: ( dom f = ND ` & f is_integrable_on M ) ;
( dom (a (#) f) = ND ` & a (#) f is_integrable_on M ) by A2, MESFUNC6:102, VALUED_1:def 5;
hence a (#) f in L1_Functions M by A1; :: thesis: verum