The following picture illustrates the idea used to define the Lebesgue integral to be like
the area under a curve.
PICT
You can see that by following the procedure illustrated in the picture and letting h
get smaller, you would expect to obtain better approximations to the area under the
curve1
although all these approximations would likely be too small. Therefore, define
∫ ∞∑
fdμ ≡ sup hμ([ih < f])
h>0i=1
Lemma 9.3.4The following inequality holds.
([ ])
∑∞ ∞∑ h- h-
h μ([ih < f]) ≤ 2μ i2 < f .
i=1 i=1
Also, it suffices to consider only h smaller than a given positive number in the abovedefinition of the integral.