A reduction technique for limit theorems in analysis and probability theory

On the best constant for the Besicovitch covering theorem

“…Reducing Besicovitch's theorem to a satellite problem 5.1 A short proof of Equation (1). The proof in [15] that Equation (1) holds (and the proof of the more general result in [6]) starts with a set A and a covering W formed by balls centered at each point of A with a finite upper bound for the radii. One then finds by a transfinite induction a well-ordered subcovering W0 = {Co, Cx, C2, ...} of W by selecting at each stage, a, a ball Ca with center not yet covered by u7<QCy and radius bigger than any competing radius divided by x.…”

“…(For a simple proof of Morse's result, see [6].) The principal use of such covering theorems in analysis is to show that "undesirable sets" are null sets with respect to some rj-finite Borel measure p .…”

“…To prove the Lebesgue Differentiation Theorem as in [6], one must show that for palmost every a G S there is a ô > 0 such that if 0 < r < ô, then /i(5(a, /•)) > i/(5(a, r)). Let A be the set of points where this is not true; fix e > 0 and a compact set K in X\S such that v(X\K) < e/ß .…”

A new proof of the two weight norm inequality for the one-sided fractional maximal operator

“…If we combine this with inequality (1) and take the limit as n tends to infinity, we get (since f is bounded) that 0 ≥ M The next lemma is due to Jesus Aldaz; the proof is in Bliedtner and Loeb [2]. …”

Submartingalas finitas, teoremas de convergencia y métodos no standard

“…Los métodos no standard han sido también abordados en [1] para establecer un resultado muy general sobre problemas límite en análisis y probabilidades, del cual se deduce en particular el teorema de convergencia para martingalas. Las ideas j. lobo expuestas en [1] son muy diferentes a las expuestas aquí, y parece más sencillo y natural, en el problema concreto de la convergencia de martingalas con métodos no standard, el recurso a las sucesiones finitas.…”