let X be non empty set ; :: thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f, g being PartFunc of X,ExtREAL st f is_integrable_on M & g is_integrable_on M holds
f + g is_integrable_on M
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f, g being PartFunc of X,ExtREAL st f is_integrable_on M & g is_integrable_on M holds
f + g is_integrable_on M
let M be sigma_Measure of S; :: thesis: for f, g being PartFunc of X,ExtREAL st f is_integrable_on M & g is_integrable_on M holds
f + g is_integrable_on M
let f, g be PartFunc of X,ExtREAL; :: thesis: ( f is_integrable_on M & g is_integrable_on M implies f + g is_integrable_on M )
assume that
A1: f is_integrable_on M and
A2: g is_integrable_on M ; :: thesis: f + g is_integrable_on M
A3: ex E2 being Element of S st
( E2 = dom g & g is E2 -measurable ) by A2;
ex E1 being Element of S st
( E1 = dom f & f is E1 -measurable ) by A1;
then ex K0 being Element of S st
( K0 = dom (f + g) & f + g is K0 -measurable ) by A3, Th47;
hence f + g is_integrable_on M by A1, A2, Lm11; :: thesis: verum