# Were there proofs of the Lebesgue Differentiation Theorem without using maximal functions?

Is there a proof of the Lebesgue Differentiation Theorem that does not involve the Hardy-Littlewood Maximal Function? For example, did Lebesgue prove it?

If there is such a proof, where can I find it?

The original proof appears in Lebesgue's Lecons sur l'intdgration et la recherche des fonctions primitives, Paris, 1904, freely available if you read French. He only considered continuous monotone functions. Young gave a proof without assuming continuity in 1911, and in 1932 Riesz gave an elementary proof for the continuous case using his "Rising Sun Lemma". In 1965 Austin gave a non-elementary geometric proof that requires intricate measure theory including the measurability of the Dini derivatives. Botsko gives a general elementary proof in American Mathematical Monthly Vol. 110, No. 9 (Nov., 2003), pp. 834-838, and according to him all other proofs use the Vitali Covering Theorem (in proving the estimate for Hardy-Littlewood Maximal Function, for instance), or the Lebesgue Density Theorem.

• +1. My answer would have been: to prove it without the maximal function, use the Vitali covering theorem. – Gerald Edgar Oct 4 '18 at 0:45