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_integrable_on M holds
f is_a.e.integrable_on M
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f being PartFunc of X,ExtREAL st f is_integrable_on M holds
f is_a.e.integrable_on M
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,ExtREAL st f is_integrable_on M holds
f is_a.e.integrable_on M
let f be PartFunc of X,ExtREAL; :: thesis: ( f is_integrable_on M implies f is_a.e.integrable_on M )
assume A1: f is_integrable_on M ; :: thesis: f is_a.e.integrable_on M
reconsider A = {} , XX = X as Element of S by MEASURE1:7;
A2: ( M . A = 0 & A c= dom f & f | ((dom f) \ A) is_integrable_on M ) by A1, VALUED_0:def 19;
then f | (A `) is_integrable_on M by Th15;
hence f is_a.e.integrable_on M by A2; :: thesis: verum