:: MESFUNC4 semantic presentation
theorem Th1: :: MESFUNC4:1
theorem Th2: :: MESFUNC4:2
theorem Th3: :: MESFUNC4:3
theorem Th4: :: MESFUNC4:4
theorem Th5: :: MESFUNC4:5
for b
1 being non
empty set for b
2 being
SigmaField of b
1for b
3 being
sigma_Measure of b
2for b
4, b
5 being
PartFunc of b
1,
ExtREAL holds
( b
4 is_simple_func_in b
2 &
dom b
4 <> {} & ( for b
6 being
set holds
( b
6 in dom b
4 implies
0. <= b
4 . b
6 ) ) & b
5 is_simple_func_in b
2 &
dom b
5 = dom b
4 & ( for b
6 being
set holds
( b
6 in dom b
5 implies
0. <= b
5 . b
6 ) ) implies ( b
4 + b
5 is_simple_func_in b
2 &
dom (b4 + b5) <> {} & ( for b
6 being
set holds
( b
6 in dom (b4 + b5) implies
0. <= (b4 + b5) . b
6 ) ) &
integral b
1,b
2,b
3,
(b4 + b5) = (integral b1,b2,b3,b4) + (integral b1,b2,b3,b5) ) )
theorem Th6: :: MESFUNC4:6