Topics In Analysis

9.3.6 Other Definitions

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,

with s_n → f and then by monotone convergence theorem,

∫ ∫ fdμ = lim sn n→∞

where if s_n

= ∑ _j=1^mc_iX_Ei,

∫ ∑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.