9.3.2 The Lebesgue Integral Nonnegative Functions
The following picture illustrates the idea used to define the Lebesgue integral to be like
the area under a curve.
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
although all these approximations would likely be too small. Therefore, define
Lemma 9.3.4 The following inequality holds.
Also, it suffices to consider only h smaller than a given positive number in the above
definition of the integral.
Let N ∈ ℕ.
Now letting N →∞ yields the claim of the lemma.
To verify the last claim, suppose M < ∫
fdμ and let δ > 0 be given. Then there exists
h > 0 such that
By the first part of this lemma,
and continuing to apply the first part,
Choose n large enough that h∕2n < δ. It follows
Since M is arbitrary, this proves the last claim.