1.6. Differentiation theorems 127 (ii) Give an alternate proof of the above claim that avoids the Lebesgue differentiation theorem. (Hint: Reduce to the case when E is bounded, then approximate E by an almost disjoint union of cubes.) (iii) Use the above result to give an alternate proof of the Steinhaus theorem (Exercise 1.6.8). Of course, one can replace cubes here by other comparable shapes, such as balls. (Indeed, a good principle to adopt in analysis is that cubes and balls are “equivalent up to constants”, in that a cube of some sidelength can be contained in a ball of comparable radius, and vice versa. This type of mental equivalence is analogous to, though not identical with, the famous dictum that a topologist cannot distinguish a doughnut from a coffee cup.) Exercise 1.6.26. (i) Give an example of a compact set K R of positive measure such that m(K I) |I| for every interval I of positive length. (Hint: First construct an open dense subset of [0, 1] of measure strictly less than 1.) (ii) Give an example of a measurable set E R such that 0 m(E I) |I| for every interval I of positive length. (Hint: First work in a bounded interval, such as (−1, 2). The complement of the set K in the first example is the union of at most countably many open intervals, thanks to Exercise 1.6.10. Now fill in these open intervals and iterate.) Exercise 1.6.27 (Approximations to the identity). Define a good kernel 15 to be a measurable function P : Rd R+ which is non-negative, radial (which means that there is a function ˜ P : [0, +∞) R+ such that P (x) = ˜(|x|)), P radially non-increasing (so that ˜ P is a non-increasing function), and has total mass Rd P (x) dx equal to 1. The functions Pt(x) := 1 td P ( x t ) for t 0 are then said to be a good family of approximations to the identity. (i) Show that the heat kernels16 Pt(x) := 1 (4πt2)d/2 e−|x|2/4t2 and Poisson kernels Pt(x) := cd t (t2+|x|2)(d+1)/2 are good families of approxima- tions to the identity, if the constant cd 0 is chosen correctly (in fact one has cd = Γ((d + 1)/2)/π(d+1)/2, but you are not required to establish this). 15Different texts have slightly different notions of what a good kernel is the “right” class of kernels to consider depends to some extent on what type of convergence results one is interested in (e.g., almost everywhere convergence, convergence in L1 or L∞ norm, etc.), and on what hypotheses one wishes to place on the original function f. 16Note that we have modified the usual formulation of the heat kernel by replacing t with t2 in order to make it conform to the notational conventions used in this exercise.
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