let X1, X2 be non empty set ; 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; 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; 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; 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; 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; 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)); ( 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
; ( 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; ( 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; verum