Chapter 26

Integrals And Derivatives26.1 The Fundamental Theorem Of Calculus

The version of the fundamental theorem of calculus found in Calculus has already beenreferred to frequently. It says that if f is a Riemann integrable function, the function

x→∫ x

af (t)dt,

has a derivative at every point where f is continuous. It is natural to ask what occurs for fin L1. It is an amazing fact that the same result is obtained aside from a set of measure zeroeven though f , being only in L1 may fail to be continuous anywhere. Proofs of this resultare based on some form of the Vitali covering theorem presented above. In what follows,the measure space is (Rn,S ,m) where m is n-dimensional Lebesgue measure although thesame theorems can be proved for arbitrary Radon measures [84]. To save notation, m iswritten in place of mn.

By Lemma 12.1.9 on Page 278 and the completeness of m, the Lebesgue measurablesets are exactly those measurable in the sense of Caratheodory. Also, to save on notation mis also the name of the outer measure defined on all of P(Rn) which is determined by mn.Recall

B(p,r) = {x : |x−p|< r}. (26.1.1)

Also define the following.

If B = B(p,r), then B̂ = B(p,5r). (26.1.2)

The first version of the Vitali covering theorem presented above will now be used toestablish the fundamental theorem of calculus. The space of locally integrable functions isthe most general one for which the maximal function defined below makes sense.

Definition 26.1.1 f ∈ L1loc(Rn) means f XB(0,R) ∈ L1(Rn) for all R > 0. For f ∈ L1

loc(Rn),the Hardy Littlewood Maximal Function, M f , is defined by

M f (x)≡ supr>0

1m(B(x,r))

∫B(x,r)

| f (y)|dy.

Theorem 26.1.2 If f ∈ L1(Rn), then for α > 0,

m([M f > α])≤ 5n

α|| f ||1.

(Here and elsewhere, [M f > α] ≡ {x ∈ Rn : M f (x) > α} with other occurrences of [ ]being defined similarly.)

Proof: Let S≡ [M f > α]. For x ∈ S, choose rx > 0 with

1m(B(x,rx))

∫B(x,rx)

| f | dm > α.

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