let X1, X2 be non empty set ; for S1 being SigmaField of X1
for S2 being SigmaField of X2
for M1 being sigma_Measure of S1
for M2 being sigma_Measure of S2
for V being Element of sigma (measurable_rectangles (S1,S2))
for A being Element of S1
for B being Element of S2 st M1 is sigma_finite & V = [:A,B:] & (product_sigma_Measure (M1,M2)) . V < +infty & M1 . A < +infty holds
sigma (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : Integral (M2,(X-vol ((E /\ V),M1))) = (product_sigma_Measure (M1,M2)) . (E /\ V) }
let S1 be SigmaField of X1; for S2 being SigmaField of X2
for M1 being sigma_Measure of S1
for M2 being sigma_Measure of S2
for V being Element of sigma (measurable_rectangles (S1,S2))
for A being Element of S1
for B being Element of S2 st M1 is sigma_finite & V = [:A,B:] & (product_sigma_Measure (M1,M2)) . V < +infty & M1 . A < +infty holds
sigma (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : Integral (M2,(X-vol ((E /\ V),M1))) = (product_sigma_Measure (M1,M2)) . (E /\ V) }
let S2 be SigmaField of X2; for M1 being sigma_Measure of S1
for M2 being sigma_Measure of S2
for V being Element of sigma (measurable_rectangles (S1,S2))
for A being Element of S1
for B being Element of S2 st M1 is sigma_finite & V = [:A,B:] & (product_sigma_Measure (M1,M2)) . V < +infty & M1 . A < +infty holds
sigma (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : Integral (M2,(X-vol ((E /\ V),M1))) = (product_sigma_Measure (M1,M2)) . (E /\ V) }
let M1 be sigma_Measure of S1; for M2 being sigma_Measure of S2
for V being Element of sigma (measurable_rectangles (S1,S2))
for A being Element of S1
for B being Element of S2 st M1 is sigma_finite & V = [:A,B:] & (product_sigma_Measure (M1,M2)) . V < +infty & M1 . A < +infty holds
sigma (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : Integral (M2,(X-vol ((E /\ V),M1))) = (product_sigma_Measure (M1,M2)) . (E /\ V) }
let M2 be sigma_Measure of S2; for V being Element of sigma (measurable_rectangles (S1,S2))
for A being Element of S1
for B being Element of S2 st M1 is sigma_finite & V = [:A,B:] & (product_sigma_Measure (M1,M2)) . V < +infty & M1 . A < +infty holds
sigma (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : Integral (M2,(X-vol ((E /\ V),M1))) = (product_sigma_Measure (M1,M2)) . (E /\ V) }
let V be Element of sigma (measurable_rectangles (S1,S2)); for A being Element of S1
for B being Element of S2 st M1 is sigma_finite & V = [:A,B:] & (product_sigma_Measure (M1,M2)) . V < +infty & M1 . A < +infty holds
sigma (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : Integral (M2,(X-vol ((E /\ V),M1))) = (product_sigma_Measure (M1,M2)) . (E /\ V) }
let A be Element of S1; for B being Element of S2 st M1 is sigma_finite & V = [:A,B:] & (product_sigma_Measure (M1,M2)) . V < +infty & M1 . A < +infty holds
sigma (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : Integral (M2,(X-vol ((E /\ V),M1))) = (product_sigma_Measure (M1,M2)) . (E /\ V) }
let B be Element of S2; ( M1 is sigma_finite & V = [:A,B:] & (product_sigma_Measure (M1,M2)) . V < +infty & M1 . A < +infty implies sigma (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : Integral (M2,(X-vol ((E /\ V),M1))) = (product_sigma_Measure (M1,M2)) . (E /\ V) } )
set K = { E where E is Element of sigma (measurable_rectangles (S1,S2)) : Integral (M2,(X-vol ((E /\ V),M1))) = (product_sigma_Measure (M1,M2)) . (E /\ V) } ;
assume that
A1:
M1 is sigma_finite
and
A2:
V = [:A,B:]
and
A3:
(product_sigma_Measure (M1,M2)) . V < +infty
and
A4:
M1 . A < +infty
; sigma (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : Integral (M2,(X-vol ((E /\ V),M1))) = (product_sigma_Measure (M1,M2)) . (E /\ V) }
A5:
{ E where E is Element of sigma (measurable_rectangles (S1,S2)) : Integral (M2,(X-vol ((E /\ V),M1))) = (product_sigma_Measure (M1,M2)) . (E /\ V) } is MonotoneClass of [:X1,X2:]
by A1, A2, A3, A4, Th113;
A6:
Field_generated_by (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : Integral (M2,(X-vol ((E /\ V),M1))) = (product_sigma_Measure (M1,M2)) . (E /\ V) }
by A1, A2, Th107;
sigma (Field_generated_by (measurable_rectangles (S1,S2))) =
sigma (DisUnion (measurable_rectangles (S1,S2)))
by SRINGS_3:22
.=
sigma (measurable_rectangles (S1,S2))
by Th1
;
hence
sigma (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : Integral (M2,(X-vol ((E /\ V),M1))) = (product_sigma_Measure (M1,M2)) . (E /\ V) }
by A5, A6, Th87; verum