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_simple_func_in S & dom f <> {} & dom f in S & M . (dom f) < +infty holds
f is_integrable_on M
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f being PartFunc of X,REAL st f is_simple_func_in S & dom f <> {} & dom f in S & M . (dom f) < +infty holds
f is_integrable_on M
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,REAL st f is_simple_func_in S & dom f <> {} & dom f in S & M . (dom f) < +infty holds
f is_integrable_on M
let f be PartFunc of X,REAL; :: thesis: ( f is_simple_func_in S & dom f <> {} & dom f in S & M . (dom f) < +infty implies f is_integrable_on M )
assume A1: f is_simple_func_in S ; :: thesis: ( not dom f <> {} or not dom f in S or not M . (dom f) < +infty or f is_integrable_on M )
then rng f is real-bounded by Lm3;
hence ( not dom f <> {} or not dom f in S or not M . (dom f) < +infty or f is_integrable_on M ) by A1, Lm4, MESFUNC6:50; :: thesis: verum