theorem Th11: :: MESFUN14:11

for a, b being Real
for A being non empty closed_interval Subset of REAL
for D being Division of A
for f being PartFunc of A,REAL st A = [.a,b.] holds
ex F being Finite_Sep_Sequence of Borel_Sets ex g being PartFunc of REAL,ExtREAL st
( dom F = dom D & union (rng F) = A & ( for k being Nat st k in dom F holds
( ( len D = 1 implies F . k = [.a,b.] ) & ( len D <> 1 implies ( ( k = 1 implies F . k = [.a,(D . k).[ ) & ( 1 < k & k < len D implies F . k = [.(D . (k -' 1)),(D . k).[ ) & ( k = len D implies F . k = [.(D . (k -' 1)),(D . k).] ) ) ) ) ) & g is_simple_func_in Borel_Sets & dom g = A & ( for x being Real st x in dom g holds
ex k being Nat st
( 1 <= k & k <= len F & x in F . k & g . x = upper_bound (rng (f | (divset (D,k)))) ) ) )