Kenneth Kuttler

11.3. THE ABSTRACT LEBESGUE INTEGRAL 237

11.3.2 The Lebesgue Integral Nonnegative FunctionsThe following picture illustrates the idea used to define the Lebesgue integral to be like thearea under a curve.

hµ([h < f ])

hµ([2h < f ])

hµ([3h < f ])

You can see that by following the procedure illustrated in the picture and letting h getsmaller, you would expect to obtain better approximations to the area under the curve1

although all these approximations would likely be too small. Therefore, define∫f dµ ≡ sup

h>0

∞

∑i=1

hµ ([ih < f ])

Lemma 11.3.4 The following inequality holds.

∞

∑i=1

hµ ([ih < f ])≤∞

∑i=1

([ih2< f])

Also, it suffices to consider only h smaller than a given positive number in the above defi-nition of the integral.

Proof:Let N ∈ N.

∑i=1

([ih2< f])

=2N

∑i=1

µ ([ih < 2 f ])

∑i=1

µ ([(2i−1)h < 2 f ])+N

∑i=1

µ ([(2i)h < 2 f ])

∑i=1

([(2i−1)

2h < f

])+

∑i=1

µ ([ih < f ])

≥N

∑i=1

µ ([ih < f ])+N

∑i=1

µ ([ih < f ]) =N

∑i=1

hµ ([ih < f ]) .

Now letting N→ ∞ yields the claim of the lemma.

1Note the difference between this picture and the one usually drawn in calculus courses where the little rect-angles are upright rather than on their sides. This illustrates a fundamental philosophical difference between theRiemann and the Lebesgue integrals. With the Riemann integral intervals are measured. With the Lebesgueintegral, it is inverse images of intervals which are measured.

11.3. THE ABSTRACT LEBESGUE INTEGRAL 23711.3.2 The Lebesgue Integral Nonnegative FunctionsThe following picture illustrates the idea used to define the Lebesgue integral to be like thearea under a curve.3hon_(|_melBh< fl)h hu((2h < f))hu([h < f])You can see that by following the procedure illustrated in the picture and letting h getsmaller, you would expect to obtain better approximations to the area under the curve!although all these approximations would likely be too small. Therefore, define[feu = sup 5 hy (ih < f])h>0 j=1Lemma 11.3.4 The following inequality holds.Eom m<E4u([i4 <4)Also, it suffices to consider only h smaller than a given positive number in the above defi-nition of the integral.Proof:LetN EN.2N 2Nm([q<s])=¥ h L (ih < 2f])- Yo Fu (((2—1)h<2f)) + +yi=1 i=1Pu ([%2n<y]) 4 +f ad (inp)N NNh. h>) se (lin< f+ se (lin < fl) = dhe ([ih < f]).i=l i=lVeNow letting N — © yields the claim of the lemma.'Note the difference between this picture and the one usually drawn in calculus courses where the little rect-angles are upright rather than on their sides. This illustrates a fundamental philosophical difference between theRiemann and the Lebesgue integrals. With the Riemann integral intervals are measured. With the Lebesgueintegral, it is inverse images of intervals which are measured.