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,REAL st f is_integrable_on M & g is_integrable_on M & f is nonnegative & g is nonnegative 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,REAL st f is_integrable_on M & g is_integrable_on M & f is nonnegative & g is nonnegative holds
f + g is_integrable_on M
let M be sigma_Measure of S; :: thesis: for f, g being PartFunc of X,REAL st f is_integrable_on M & g is_integrable_on M & f is nonnegative & g is nonnegative holds
f + g is_integrable_on M
let f, g be PartFunc of X,REAL; :: thesis: ( f is_integrable_on M & g is_integrable_on M & f is nonnegative & g is nonnegative implies f + g is_integrable_on M )
assume that
A1: ( f is_integrable_on M & g is_integrable_on M ) and
A2: ( f is nonnegative & g is nonnegative ) ; :: thesis: f + g is_integrable_on M
( R_EAL f is_integrable_on M & R_EAL g is_integrable_on M ) by A1;
then (R_EAL f) + (R_EAL g) is_integrable_on M by A2, MESFUNC5:106;
then R_EAL (f + g) is_integrable_on M by Th23;
hence f + g is_integrable_on M ; :: thesis: verum