To review and summarize the above, if f ≥ 0 is measurable,
∫ ∞∑
fdμ ≡ sup hμ([f > ih]) (9.3.20)
h>0i=1
(9.3.20)
another way to get the same thing for ∫fdμ is to take an increasing sequence of
nonnegative simple functions,
{sn}
with sn
(ω)
→ f
(ω)
and then by monotone
convergence theorem,
∫ ∫
fdμ = lim sn
n→∞
where if sn
(ω )
= ∑j=1mciXEi
(ω)
,
∫ ∑m
sndμ = cim (Ei) .
i=1
Similarly this also shows that for such nonnegative measurable function,
∫ {∫ }
fdμ = sup s : 0 ≤ s ≤ f, s simple
which is the usual way of defining the Lebesgue integral for nonnegative simple functions
in most books. I have done it differently because this approach led to an easier proof of
the Monotone convergence theorem. Here is an equivalent definition of the integral. The
fact it is well defined has been discussed above.
Definition 9.3.14For s a nonnegative simple function,
n ∫ n
s (ω ) = ∑ ckXE (ω ), s = ∑ ckμ(Ek) .
k=1 k k=1