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 B being Element of S1 st M1 . B < +infty holds
sigma (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : ex F being Function of X2,ExtREAL st
( ( for y being Element of X2 holds F . y = M1 . ((Measurable-Y-section (E,y)) /\ B) ) & ( for V being Element of S2 holds F is V -measurable ) ) }
let S1 be SigmaField of X1; :: thesis: for S2 being SigmaField of X2
for M1 being sigma_Measure of S1
for B being Element of S1 st M1 . B < +infty holds
sigma (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : ex F being Function of X2,ExtREAL st
( ( for y being Element of X2 holds F . y = M1 . ((Measurable-Y-section (E,y)) /\ B) ) & ( for V being Element of S2 holds F is V -measurable ) ) }
let S2 be SigmaField of X2; :: thesis: for M1 being sigma_Measure of S1
for B being Element of S1 st M1 . B < +infty holds
sigma (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : ex F being Function of X2,ExtREAL st
( ( for y being Element of X2 holds F . y = M1 . ((Measurable-Y-section (E,y)) /\ B) ) & ( for V being Element of S2 holds F is V -measurable ) ) }
let M1 be sigma_Measure of S1; :: thesis: for B being Element of S1 st M1 . B < +infty holds
sigma (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : ex F being Function of X2,ExtREAL st
( ( for y being Element of X2 holds F . y = M1 . ((Measurable-Y-section (E,y)) /\ B) ) & ( for V being Element of S2 holds F is V -measurable ) ) }
let B be Element of S1; :: thesis: ( M1 . B < +infty implies sigma (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : ex F being Function of X2,ExtREAL st
( ( for y being Element of X2 holds F . y = M1 . ((Measurable-Y-section (E,y)) /\ B) ) & ( for V being Element of S2 holds F is V -measurable ) ) } )
set K = { E where E is Element of sigma (measurable_rectangles (S1,S2)) : ex F being Function of X2,ExtREAL st
( ( for y being Element of X2 holds F . y = M1 . ((Measurable-Y-section (E,y)) /\ B) ) & ( for V being Element of S2 holds F is V -measurable ) ) } ;
assume M1 . B < +infty ; :: thesis: sigma (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : ex F being Function of X2,ExtREAL st
( ( for y being Element of X2 holds F . y = M1 . ((Measurable-Y-section (E,y)) /\ B) ) & ( for V being Element of S2 holds F is V -measurable ) ) }
then A1: { E where E is Element of sigma (measurable_rectangles (S1,S2)) : ex F being Function of X2,ExtREAL st
( ( for y being Element of X2 holds F . y = M1 . ((Measurable-Y-section (E,y)) /\ B) ) & ( for V being Element of S2 holds F is V -measurable ) ) } is MonotoneClass of [:X1,X2:] by Th85;
A2: Field_generated_by (measurable_rectangles (S1,S2)) c= { E where E is Element of sigma (measurable_rectangles (S1,S2)) : ex F being Function of X2,ExtREAL st
( ( for y being Element of X2 holds F . y = M1 . ((Measurable-Y-section (E,y)) /\ B) ) & ( for V being Element of S2 holds F is V -measurable ) ) } by Th81;
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)) : ex F being Function of X2,ExtREAL st
( ( for y being Element of X2 holds F . y = M1 . ((Measurable-Y-section (E,y)) /\ B) ) & ( for V being Element of S2 holds F is V -measurable ) ) } by A1, A2, Th87; :: thesis: verum