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_integrable_on M holds
( -infty < Integral (M,f) & Integral (M,f) < +infty )
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f being PartFunc of X,REAL st f is_integrable_on M holds
( -infty < Integral (M,f) & Integral (M,f) < +infty )
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,REAL st f is_integrable_on M holds
( -infty < Integral (M,f) & Integral (M,f) < +infty )
let f be PartFunc of X,REAL; :: thesis: ( f is_integrable_on M implies ( -infty < Integral (M,f) & Integral (M,f) < +infty ) )
assume f is_integrable_on M ; :: thesis: ( -infty < Integral (M,f) & Integral (M,f) < +infty )
then R_EAL f is_integrable_on M by Def9;
hence ( -infty < Integral (M,f) & Integral (M,f) < +infty ) by MESFUNC5:96; :: thesis: verum