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

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