let X1, X2 be non empty set ; :: thesis: for S1 being SigmaField of X1
for S2 being SigmaField of X2
for f being PartFunc of [:X1,X2:],ExtREAL
for x being Element of X1
for y being Element of X2
for E being Element of sigma (measurable_rectangles (S1,S2)) st E c= dom f & f is E -measurable holds
( ProjPMap1 ((max+ f),x) is Measurable-X-section (E,x) -measurable & ProjPMap2 ((max+ f),y) is Measurable-Y-section (E,y) -measurable & ProjPMap1 ((max- f),x) is Measurable-X-section (E,x) -measurable & ProjPMap2 ((max- f),y) is Measurable-Y-section (E,y) -measurable )
let S1 be SigmaField of X1; :: thesis: for S2 being SigmaField of X2
for f being PartFunc of [:X1,X2:],ExtREAL
for x being Element of X1
for y being Element of X2
for E being Element of sigma (measurable_rectangles (S1,S2)) st E c= dom f & f is E -measurable holds
( ProjPMap1 ((max+ f),x) is Measurable-X-section (E,x) -measurable & ProjPMap2 ((max+ f),y) is Measurable-Y-section (E,y) -measurable & ProjPMap1 ((max- f),x) is Measurable-X-section (E,x) -measurable & ProjPMap2 ((max- f),y) is Measurable-Y-section (E,y) -measurable )
let S2 be SigmaField of X2; :: thesis: for f being PartFunc of [:X1,X2:],ExtREAL
for x being Element of X1
for y being Element of X2
for E being Element of sigma (measurable_rectangles (S1,S2)) st E c= dom f & f is E -measurable holds
( ProjPMap1 ((max+ f),x) is Measurable-X-section (E,x) -measurable & ProjPMap2 ((max+ f),y) is Measurable-Y-section (E,y) -measurable & ProjPMap1 ((max- f),x) is Measurable-X-section (E,x) -measurable & ProjPMap2 ((max- f),y) is Measurable-Y-section (E,y) -measurable )
let f be PartFunc of [:X1,X2:],ExtREAL; :: thesis: for x being Element of X1
for y being Element of X2
for E being Element of sigma (measurable_rectangles (S1,S2)) st E c= dom f & f is E -measurable holds
( ProjPMap1 ((max+ f),x) is Measurable-X-section (E,x) -measurable & ProjPMap2 ((max+ f),y) is Measurable-Y-section (E,y) -measurable & ProjPMap1 ((max- f),x) is Measurable-X-section (E,x) -measurable & ProjPMap2 ((max- f),y) is Measurable-Y-section (E,y) -measurable )
let x be Element of X1; :: thesis: for y being Element of X2
for E being Element of sigma (measurable_rectangles (S1,S2)) st E c= dom f & f is E -measurable holds
( ProjPMap1 ((max+ f),x) is Measurable-X-section (E,x) -measurable & ProjPMap2 ((max+ f),y) is Measurable-Y-section (E,y) -measurable & ProjPMap1 ((max- f),x) is Measurable-X-section (E,x) -measurable & ProjPMap2 ((max- f),y) is Measurable-Y-section (E,y) -measurable )
let y be Element of X2; :: thesis: for E being Element of sigma (measurable_rectangles (S1,S2)) st E c= dom f & f is E -measurable holds
( ProjPMap1 ((max+ f),x) is Measurable-X-section (E,x) -measurable & ProjPMap2 ((max+ f),y) is Measurable-Y-section (E,y) -measurable & ProjPMap1 ((max- f),x) is Measurable-X-section (E,x) -measurable & ProjPMap2 ((max- f),y) is Measurable-Y-section (E,y) -measurable )
let A be Element of sigma (measurable_rectangles (S1,S2)); :: thesis: ( A c= dom f & f is A -measurable implies ( ProjPMap1 ((max+ f),x) is Measurable-X-section (A,x) -measurable & ProjPMap2 ((max+ f),y) is Measurable-Y-section (A,y) -measurable & ProjPMap1 ((max- f),x) is Measurable-X-section (A,x) -measurable & ProjPMap2 ((max- f),y) is Measurable-Y-section (A,y) -measurable ) )
assume that
A1: A c= dom f and
A2: f is A -measurable ; :: thesis: ( ProjPMap1 ((max+ f),x) is Measurable-X-section (A,x) -measurable & ProjPMap2 ((max+ f),y) is Measurable-Y-section (A,y) -measurable & ProjPMap1 ((max- f),x) is Measurable-X-section (A,x) -measurable & ProjPMap2 ((max- f),y) is Measurable-Y-section (A,y) -measurable )
A3: ( max+ f is nonnegative & max- f is nonnegative ) by MESFUN11:5;
A4: max+ f is A -measurable by A2, MESFUNC2:25;
A5: max- f is A -measurable by A1, A2, MESFUNC2:26;
dom (max+ f) = dom f by MESFUNC2:def 2;
hence ( ProjPMap1 ((max+ f),x) is Measurable-X-section (A,x) -measurable & ProjPMap2 ((max+ f),y) is Measurable-Y-section (A,y) -measurable ) by A1, A3, A4, Lm3; :: thesis: ( ProjPMap1 ((max- f),x) is Measurable-X-section (A,x) -measurable & ProjPMap2 ((max- f),y) is Measurable-Y-section (A,y) -measurable )
dom (max- f) = dom f by MESFUNC2:def 3;
hence ( ProjPMap1 ((max- f),x) is Measurable-X-section (A,x) -measurable & ProjPMap2 ((max- f),y) is Measurable-Y-section (A,y) -measurable ) by A1, A3, A5, Lm3; :: thesis: verum