let X1, X2 be non empty set ; :: thesis: 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 M2 is sigma_finite & V = [:A,B:] & (product_sigma_Measure (M1,M2)) . V < +infty & M2 . B < +infty holds
sigma (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : Integral (M1,(Y-vol ((E /\ V),M2))) = (product_sigma_Measure (M1,M2)) . (E /\ V) }
let S1 be SigmaField of X1; :: thesis: 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 M2 is sigma_finite & V = [:A,B:] & (product_sigma_Measure (M1,M2)) . V < +infty & M2 . B < +infty holds
sigma (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : Integral (M1,(Y-vol ((E /\ V),M2))) = (product_sigma_Measure (M1,M2)) . (E /\ V) }
let S2 be SigmaField of X2; :: thesis: 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 M2 is sigma_finite & V = [:A,B:] & (product_sigma_Measure (M1,M2)) . V < +infty & M2 . B < +infty holds
sigma (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : Integral (M1,(Y-vol ((E /\ V),M2))) = (product_sigma_Measure (M1,M2)) . (E /\ V) }
let M1 be sigma_Measure of S1; :: thesis: 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 M2 is sigma_finite & V = [:A,B:] & (product_sigma_Measure (M1,M2)) . V < +infty & M2 . B < +infty holds
sigma (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : Integral (M1,(Y-vol ((E /\ V),M2))) = (product_sigma_Measure (M1,M2)) . (E /\ V) }
let M2 be sigma_Measure of S2; :: thesis: for V being Element of sigma (measurable_rectangles (S1,S2))
for A being Element of S1
for B being Element of S2 st M2 is sigma_finite & V = [:A,B:] & (product_sigma_Measure (M1,M2)) . V < +infty & M2 . B < +infty holds
sigma (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : Integral (M1,(Y-vol ((E /\ V),M2))) = (product_sigma_Measure (M1,M2)) . (E /\ V) }
let V be Element of sigma (measurable_rectangles (S1,S2)); :: thesis: for A being Element of S1
for B being Element of S2 st M2 is sigma_finite & V = [:A,B:] & (product_sigma_Measure (M1,M2)) . V < +infty & M2 . B < +infty holds
sigma (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : Integral (M1,(Y-vol ((E /\ V),M2))) = (product_sigma_Measure (M1,M2)) . (E /\ V) }
let A be Element of S1; :: thesis: for B being Element of S2 st M2 is sigma_finite & V = [:A,B:] & (product_sigma_Measure (M1,M2)) . V < +infty & M2 . B < +infty holds
sigma (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : Integral (M1,(Y-vol ((E /\ V),M2))) = (product_sigma_Measure (M1,M2)) . (E /\ V) }
let B be Element of S2; :: thesis: ( M2 is sigma_finite & V = [:A,B:] & (product_sigma_Measure (M1,M2)) . V < +infty & M2 . B < +infty implies sigma (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : Integral (M1,(Y-vol ((E /\ V),M2))) = (product_sigma_Measure (M1,M2)) . (E /\ V) } )
set K = { E where E is Element of sigma (measurable_rectangles (S1,S2)) : Integral (M1,(Y-vol ((E /\ V),M2))) = (product_sigma_Measure (M1,M2)) . (E /\ V) } ;
assume that
A1: M2 is sigma_finite and
A2: V = [:A,B:] and
A3: (product_sigma_Measure (M1,M2)) . V < +infty and
A4: M2 . B < +infty ; :: thesis: sigma (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : Integral (M1,(Y-vol ((E /\ V),M2))) = (product_sigma_Measure (M1,M2)) . (E /\ V) }
A5: { E where E is Element of sigma (measurable_rectangles (S1,S2)) : Integral (M1,(Y-vol ((E /\ V),M2))) = (product_sigma_Measure (M1,M2)) . (E /\ V) } is MonotoneClass of [:X1,X2:] by A1, A2, A3, A4, Th112;
A6: Field_generated_by (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : Integral (M1,(Y-vol ((E /\ V),M2))) = (product_sigma_Measure (M1,M2)) . (E /\ V) } by A1, A2, Th106;
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 (M1,(Y-vol ((E /\ V),M2))) = (product_sigma_Measure (M1,M2)) . (E /\ V) } by A5, A6, Th87; :: thesis: verum