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,ExtREAL
for E, A being Element of S st f is nonnegative & E = dom f & f is E -measurable & M . A = 0 holds
integral+ (M,(f | (E \ A))) = integral+ (M,f)
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f being PartFunc of X,ExtREAL
for E, A being Element of S st f is nonnegative & E = dom f & f is E -measurable & M . A = 0 holds
integral+ (M,(f | (E \ A))) = integral+ (M,f)
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,ExtREAL
for E, A being Element of S st f is nonnegative & E = dom f & f is E -measurable & M . A = 0 holds
integral+ (M,(f | (E \ A))) = integral+ (M,f)
let f be PartFunc of X,ExtREAL; :: thesis: for E, A being Element of S st f is nonnegative & E = dom f & f is E -measurable & M . A = 0 holds
integral+ (M,(f | (E \ A))) = integral+ (M,f)
let E, A be Element of S; :: thesis: ( f is nonnegative & E = dom f & f is E -measurable & M . A = 0 implies integral+ (M,(f | (E \ A))) = integral+ (M,f) )
assume that
A1: f is nonnegative and
A2: E = dom f and
A3: f is E -measurable and
A4: M . A = 0 ; :: thesis: integral+ (M,(f | (E \ A))) = integral+ (M,f)
set B = E \ A;
A \/ (E \ A) = A \/ E by XBOOLE_1:39;
then A5: dom f = (dom f) /\ (A \/ (E \ A)) by A2, XBOOLE_1:7, XBOOLE_1:28
.= dom (f | (A \/ (E \ A))) by RELAT_1:61 ;
for x being object st x in dom (f | (A \/ (E \ A))) holds
(f | (A \/ (E \ A))) . x = f . x by FUNCT_1:47;
then A6: f | (A \/ (E \ A)) = f by A5, FUNCT_1:2;
integral+ (M,(f | (A \/ (E \ A)))) = (integral+ (M,(f | A))) + (integral+ (M,(f | (E \ A)))) by A1, A2, A3, Th81, XBOOLE_1:79;
then integral+ (M,f) = 0. + (integral+ (M,(f | (E \ A)))) by A1, A2, A3, A4, A6, Th82;
hence integral+ (M,(f | (E \ A))) = integral+ (M,f) by XXREAL_3:4; :: thesis: verum