124 1. Measure theory that has an infinite run time, unless one has a suitably strong convergence result for the algorithm that allows one to take limits, either in the classical sense or in the more general sense of jumping to limit ordinals in the latter case, one needs to use transfinite induction in order to ensure that the use of such algorithms is rigorous see §2.4 of An epsilon of room, Vol. I.) Remark 1.6.23. The actual Vitali covering lemma [Vi1908] is slightly different than this one, but we will not need it here. Actually, there is a family of related covering lemmas which are useful for a variety of tasks in harmonic analysis see, for instance, [deG1981] for further discussion. Now we can prove the Hardy-Littlewood inequality, which we will do with the constant Cd := 3d. It suffices to verify the claim with strict in- equality, m({x Rd : sup r0 1 m(B(x, r)) B(x,r) |f(y)| dy λ}) Cd λ R |f(t)| dt, as the non-strict case then follows by perturbing λ slightly and then taking limits. Fix f and λ. By inner regularity, it suffices to show that m(K) 3d λ R |f(t)| dt whenever K is a compact set that is contained in {x Rd : sup r0 1 m(B(x, r)) B(x,r) |f(y)| dy λ}. By construction, for every x K, there exists an open ball B(x, r) such that (1.28) 1 m(B(x, r)) B(x,r) |f(y)| dy λ. By compactness of K, we can cover K by a finite number B1,...,Bn of such balls. Applying the Vitali-type covering lemma, we can find a subcollection B1,...,Bm of disjoint balls such that m( n i=1 Bi) 3d m j=1 m(Bj). By (1.28), on each ball Bj we have m(Bj) 1 λ B j |f(y)| dy
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