set V = { f where f is PartFunc of X,REAL : ex Ef being Element of S st
( M . (Ef `) = 0 & dom f = Ef & f is Ef -measurable & (abs f) to_power k is_integrable_on M ) } ;
A1: { f where f is PartFunc of X,REAL : ex Ef being Element of S st
( M . (Ef `) = 0 & dom f = Ef & f is Ef -measurable & (abs f) to_power k is_integrable_on M ) } c= PFuncs (X,REAL) reconsider g = X --> (In (0,REAL)) as Function of X,REAL by FUNCOP_1:46;
reconsider Ef = X as Element of S by MEASURE1:34;
set h = (abs g) to_power k;
A2: dom g = X by FUNCOP_1:13;
for x being set st x in dom g holds
g . x = 0 by FUNCOP_1:7;
then A3: g is Ef -measurable by A2, LPSPACE1:52;
Ef ` = {} by XBOOLE_1:37;
then A4: M . (Ef `) = 0 by VALUED_0:def 19;
for x being object st x in dom (X --> 0) holds
0 <= (X --> 0) . x ;
then abs g = X --> 0 by Th14, MESFUNC6:52;
then A5: (abs g) to_power k = g by Th12;
then for x being Element of X st x in dom ((abs g) to_power k) holds
((abs g) to_power k) . x = 0 by FUNCOP_1:7;
then (abs g) to_power k is_integrable_on M by Th15, A5, A2;
then g in { f where f is PartFunc of X,REAL : ex Ef being Element of S st
( M . (Ef `) = 0 & dom f = Ef & f is Ef -measurable & (abs f) to_power k is_integrable_on M ) } by A3, A4, A2;
hence { f where f is PartFunc of X,REAL : ex Ef being Element of S st
( M . (Ef `) = 0 & dom f = Ef & f is Ef -measurable & (abs f) to_power k is_integrable_on M ) } is non empty Subset of (RLSp_PFunct X) by A1; :: thesis: verum