116 1. Measure theory no explicit bound on the rate of convergence. For instance, in Proposi- tion 1.6.13, we know that for any ε 0, there exists δ 0 such that Rd |fh(x) − f(x)| dx ≤ ε whenever |h| ≤ δ, but we do not know exactly how δ depends on ε and f. Actually, the proof does eventually give such a bound, but it depends on “how measurable” the function f is, or more precisely how “easy” it is to approximate f by a “nice” function. To il- lustrate this issue, let’s work in one dimension and consider the function f(x) := sin(Nx)1[0,2π](x), where N ≥ 1 is a large integer. On the one hand, f is bounded in the L1 norm uniformly in N: R |f(x)| dx ≤ 2π (indeed, the left-hand side is equal to 2). On the other hand, it is not hard to see that R |fπ/N (x) − f(x)| dx ≥ c for some absolute constant c 0. Thus, if one force R |fh(x) − f(x)| dx to drop below c, one has to make h at most π/N from the origin. Making N large, we thus see that the rate of convergence of R |fh(x) − f(x)| dx to zero can be arbitrarily slow, even though f is bounded in L1. The problem is that as N gets large, it becomes increasingly diﬃcult to approximate f well by a “nice” function, by which we mean a uniformly continuous function with a reasonable modulus of con- tinuity, due to the increasingly oscillatory nature of f. See [Ta2008, §1.4] for some further discussion of this issue, and what quantitative substitutes are available for such qualitative results. Now we return to the Lebesgue differentiation theorem, and apply the density argument. The dense subclass result is already contained in Corol- lary 1.6.10, which asserts that (1.20) holds for all continuous functions f. The quantitative estimate we will need is the following special case of the Hardy-Littlewood maximal inequality: Lemma 1.6.16 (One-sided Hardy-Littlewood maximal inequality). Let f : R → C be an absolutely integrable function, and let λ 0. Then m({x ∈ R : sup h0 1 h [x,x+h] |f(t)| dt ≥ λ}) ≤ 1 λ R |f(t)| dt. We will prove this lemma shortly, but let us first see how this, combined with the dense subclass result, will give the Lebesgue differentiation theorem. Let f : R → C be absolutely integrable, and let ε, λ 0 be arbitrary. Then by Littlewood’s second principle, we can find a function g : R → C which is continuous and compactly supported, with R |f(x) − g(x)| dx ≤ ε. Applying the one-sided Hardy-Littlewood maximal inequality, we conclude that m({x ∈ R : sup h0 1 h [x,x+h] |f(t) − g(t)| dt ≥ λ}) ≤ ε λ .

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