let X1, X2 be non empty set ; for S1 being SigmaField of X1
for S2 being SigmaField of X2
for M2 being sigma_Measure of S2
for x being Element of X1
for E being Element of sigma (measurable_rectangles (S1,S2))
for r being Real st M2 is sigma_finite holds
( (r (#) (Y-vol (E,M2))) . x = Integral (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) & ( r >= 0 implies (r (#) (Y-vol (E,M2))) . x = integral+ (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) ) )
let S1 be SigmaField of X1; for S2 being SigmaField of X2
for M2 being sigma_Measure of S2
for x being Element of X1
for E being Element of sigma (measurable_rectangles (S1,S2))
for r being Real st M2 is sigma_finite holds
( (r (#) (Y-vol (E,M2))) . x = Integral (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) & ( r >= 0 implies (r (#) (Y-vol (E,M2))) . x = integral+ (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) ) )
let S2 be SigmaField of X2; for M2 being sigma_Measure of S2
for x being Element of X1
for E being Element of sigma (measurable_rectangles (S1,S2))
for r being Real st M2 is sigma_finite holds
( (r (#) (Y-vol (E,M2))) . x = Integral (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) & ( r >= 0 implies (r (#) (Y-vol (E,M2))) . x = integral+ (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) ) )
let M2 be sigma_Measure of S2; for x being Element of X1
for E being Element of sigma (measurable_rectangles (S1,S2))
for r being Real st M2 is sigma_finite holds
( (r (#) (Y-vol (E,M2))) . x = Integral (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) & ( r >= 0 implies (r (#) (Y-vol (E,M2))) . x = integral+ (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) ) )
let x be Element of X1; for E being Element of sigma (measurable_rectangles (S1,S2))
for r being Real st M2 is sigma_finite holds
( (r (#) (Y-vol (E,M2))) . x = Integral (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) & ( r >= 0 implies (r (#) (Y-vol (E,M2))) . x = integral+ (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) ) )
let E be Element of sigma (measurable_rectangles (S1,S2)); for r being Real st M2 is sigma_finite holds
( (r (#) (Y-vol (E,M2))) . x = Integral (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) & ( r >= 0 implies (r (#) (Y-vol (E,M2))) . x = integral+ (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) ) )
let r be Real; ( M2 is sigma_finite implies ( (r (#) (Y-vol (E,M2))) . x = Integral (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) & ( r >= 0 implies (r (#) (Y-vol (E,M2))) . x = integral+ (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) ) ) )
assume A1:
M2 is sigma_finite
; ( (r (#) (Y-vol (E,M2))) . x = Integral (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) & ( r >= 0 implies (r (#) (Y-vol (E,M2))) . x = integral+ (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) ) )
set p2 = ProjPMap1 ((chi (E,[:X1,X2:])),x);
chi (r,E,[:X1,X2:]) = r (#) (chi (E,[:X1,X2:]))
by Th1;
then A2:
ProjPMap1 ((chi (r,E,[:X1,X2:])),x) = r (#) (ProjPMap1 ((chi (E,[:X1,X2:])),x))
by Th29;
A3:
ProjPMap1 ((chi (E,[:X1,X2:])),x) is nonnegative
by Th32;
A4:
dom (r (#) (Y-vol (E,M2))) = X1
by FUNCT_2:def 1;
A5:
chi (E,[:X1,X2:]) is_simple_func_in sigma (measurable_rectangles (S1,S2))
by Th12;
then Integral (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) =
r * (integral' (M2,(ProjPMap1 ((chi (E,[:X1,X2:])),x))))
by A2, A3, Th31, MESFUN11:59
.=
r * ((Y-vol (E,M2)) . x)
by A1, Th52
;
hence A7:
(r (#) (Y-vol (E,M2))) . x = Integral (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x)))
by A4, MESFUNC1:def 6; ( r >= 0 implies (r (#) (Y-vol (E,M2))) . x = integral+ (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) )
thus
( r >= 0 implies (r (#) (Y-vol (E,M2))) . x = integral+ (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x))) )
verumproof
assume
r >= 0
;
(r (#) (Y-vol (E,M2))) . x = integral+ (M2,(ProjPMap1 ((chi (r,E,[:X1,X2:])),x)))
then A8:
r (#) (ProjPMap1 ((chi (E,[:X1,X2:])),x)) is
nonnegative
by A3, MESFUNC5:20;
r (#) (ProjPMap1 ((chi (E,[:X1,X2:])),x)) is_simple_func_in S2
by A5, Th31, MESFUNC5:39;
hence
(r (#) (Y-vol (E,M2))) . x = integral+ (
M2,
(ProjPMap1 ((chi (r,E,[:X1,X2:])),x)))
by A2, A7, A8, MESFUNC5:89;
verum
end;