1
Q
f: X –> R is an upper function if
….
A
there is a sequence of step functions
fn ↑ f a.e. (pointwise)
and lim ∫ fn < ∞
2
Q
are upper functions measurable?
A
yes!
because step functions are measurable
and lim step is measurable!
3
Q
careful:
f upper doesn’t mean -f is upper
A
4
Q
f, g upper implies…
A
f + g, cf, max{f, g}
are upper
and
∫ f + g = ∫ f + ∫ g,
∫cf = c ∫f
5
Q
If f,g upper and f ≥ g a.e., then …
A
∫ f ≥ ∫ g
(same if g = 0)
6
Q
If fn is the generating sequence of f, then …
A
∫ f = lim ∫ fn
7
Q
If fn ↓ 0 a.e., then …
A
∫ fn ↓ 0
8
Q
a set A is relatively open in E if …
A
A = E n Open set in X
