let X be non empty set ; for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,ExtREAL
for A, B being Element of S st f is_integrable_on M & B = (dom f) \ A holds
( f | A is_integrable_on M & Integral (M,f) = (Integral (M,(f | A))) + (Integral (M,(f | B))) )
let S be SigmaField of X; for M being sigma_Measure of S
for f being PartFunc of X,ExtREAL
for A, B being Element of S st f is_integrable_on M & B = (dom f) \ A holds
( f | A is_integrable_on M & Integral (M,f) = (Integral (M,(f | A))) + (Integral (M,(f | B))) )
let M be sigma_Measure of S; for f being PartFunc of X,ExtREAL
for A, B being Element of S st f is_integrable_on M & B = (dom f) \ A holds
( f | A is_integrable_on M & Integral (M,f) = (Integral (M,(f | A))) + (Integral (M,(f | B))) )
let f be PartFunc of X,ExtREAL; for A, B being Element of S st f is_integrable_on M & B = (dom f) \ A holds
( f | A is_integrable_on M & Integral (M,f) = (Integral (M,(f | A))) + (Integral (M,(f | B))) )
let A, B be Element of S; ( f is_integrable_on M & B = (dom f) \ A implies ( f | A is_integrable_on M & Integral (M,f) = (Integral (M,(f | A))) + (Integral (M,(f | B))) ) )
assume that
A1:
f is_integrable_on M
and
A2:
B = (dom f) \ A
; ( f | A is_integrable_on M & Integral (M,f) = (Integral (M,(f | A))) + (Integral (M,(f | B))) )
A \/ B = A \/ (dom f)
by A2, XBOOLE_1:39;
then A3:
(dom f) /\ (A \/ B) = dom f
by XBOOLE_1:7, XBOOLE_1:28;
A4: f | (A \/ B) =
(f | (dom f)) | (A \/ B)
by GRFUNC_1:23
.=
f | ((dom f) /\ (A \/ B))
by RELAT_1:71
.=
f
by A3, GRFUNC_1:23
;
A misses B
by A2, XBOOLE_1:79;
hence
( f | A is_integrable_on M & Integral (M,f) = (Integral (M,(f | A))) + (Integral (M,(f | B))) )
by A1, A4, Th97, Th98; verum