INT. J. MATH. EDUC. SCI. TECHNOL., 1988, VOL. 19, NO. 4, 607-609

An alternative proof of Vitali's covering theorem

by A. J. B. WARD

School of Mathematics, Kingston Polytechnic, Penrhyn Road,Kingston-upon-Thames, Surrey KT1 2EE, England

(Received 3 March 1986)

In the teaching of the theory of the Lebesgue integral to third yearundergraduates, it has been found that some of the theorems present considerabledifficulty to the student. One of these 'difficult' theorems is Vitali's coveringtheorem and the following simpler proof is offered as an alternative to the moreusual proof due to Banach.

1. IntroductionThe theory of the Lebesgue integral is of great importance in many branches of

both pure and applied mathematics. For instance, the full theory of trigonometricseries ('Fourier series') cannot properly be presented without it. Unfortunately,many of the theorems involved require proofs which are not easy for the averagestudent to understand. One such theorem is Vitali's covering theorem which isessential for some of the deeper results in the theory of the indefinite integral. Theusual proof offered is due to Banach (see [l]p.46, [2] p. 85 or [3] p. 69) but the authorhas found that the proof to be presented here is generally found more acceptable tothe student, being more 'intuitive' (though equally rigorous). In fact it is probablybest to offer both proofs and allow the student to choose the one which he prefers. Aswill be seen, the two proofs are very different.

2. Definitions and lemmasIn what follows all sets are subsets of IR, the set of all real numbers. The notation

^E is used to denote (R — E), the complement of the set E with respect to R. Also, rhEdenotes the Lebesgue outer measure of E, while mE denotes the Lebesgue measureof the (measurable) set E. We give two lemmas.

Lemma 1 Suppose £ c R and let a be a finite set of real closed intervals{IuI2,...In}.Let

Uk=l

Then we can find a subset {Jlt J2, • • -Jm}, m^.n, of a such that the Jk are pairwisedisjoint and

m

covers a part of E of outer measure at least \mE.

608 A. J. B. Ward

Proof We may assume, without loss of generality, that

i,t 0 hk = l

for any/) in the range 1,2,.. . n; since any Ip which is contained in the union of the restof the Ik may be suppressed without altering the fact that the union of the Ik covers E.(In fact it is possible to lay down a precise law to determine which of the Ik are to besuppressed.) We may further assume that the intervals Ik are numbered in ascendingorder of their left endpoints, i.e. if Ik

= {.ak>bk\, (k—\...n), then al <a2<- • • <«„•Now write cxl = {Il,I3,J5,I1,...} and a2 = {I2>14.. 16. • • •} a n d let the unions of theintervals of al and <r2 be denoted respectively by S1 and S2- Then clearly theintervals of a1 are pairwise disjoint and the intervals of a2 are pairwise disjoint andone or other of S1, S2 covers a part of E of outer measure at least jihE (see the figure).

Lemma 2 Suppose E c (R and rhE < 00. Let E be contained in the union [j I of an

infinity of closed intervals / which are members of a family a. Then E can be coveredby an enumerable infinity of the intervals / of a, i.e.

where each Ikeo~. The intervals Ik may overlap.The proof of lemma 2 is omitted, since it is really a variation of Lindelof's

theorem (see [1] p. 7 or [2] p. 26) and follows (almost) at once from that theorem.

3. Vitali's theoremWe first give the following definition: suppose Ea U and let a be a set of closed

real intervals such that, given any xeE and any real £>0, 3/ecr such that xel andml<e. Then the set a of intervals / is said to form a Vitali covering of E.

Vitali's theorem Let a be a set of closed real intervals / forming a Vitali coveringof E where rhE < 00. Then an enumerable infinity of pairwise disjoint intervals of awill cover E almost everywhere.

Proof We can cover a subset of E of outer measure at least \mE by a finitenumber of (possibily overlapping) intervals of a by lemma 2. Then, using lemma 1,we can cover a part of E of outer measure at least \mE by a finite number of pairwisedisjoint intervals of a. Thus we can find E* c E with

E*cS= \J Ik and mE*^\mEk=l

where each Ik{k= 1 . . . ri) belongs to a, the Ik being pairwise disjoint.We next note that (E—E*) has a Vitali covering by intervals of a all lying in <^5

and so, by repeating the above argument, we can cover a part of (E—E*) of outermeasure at least \rh(E—E*) by a finite number of pairwise disjoint intervals of a alllying in &S. We can repeat the above argument as many times as we please and thepart of E not covered at the nth stage has outer measure at most (f )"ihE—>0 as n—>co.

Alternative proof of Vitali's covering theorem 609

References[1] BURKILL, J. C, 1953, The Lebesgue Integral, Cambridge Tracts in Mathematics and

Mathematical Physics, no. 40 (Cambridge University Press).[2] KESTLEMAN, H., 1960, Modern Theories of Integration (New York: Dover).[3] ROGOSINSKI, W. W., 1952, Volume and Integral (Edinburgh, London: Oliver and Boyd).