Rising sun lemma

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Rising sun lemma.svg

In mathematical analysis, the rising sun lemma is a lemma due to Frigyes Riesz, used in the proof of the Hardy–Littlewood maximal theorem. The lemma was a precursor in one dimension of the Calderón–Zygmund lemma. [1]

The lemma is stated as follows:[2]

Let g(x) be a real-valued continuous function on the interval [a,b], and let E be the set of x ∈ (a,b) such that g(y) > g(x) for some y with x < y < b.
Then E is an open set, and can be written as a disjoint union of intervals
E=\bigcup_k (a_k,b_k)
such that g(ak) = g(bk), except possibly if ak = a when g(bk) ≥ g(ak).

The colorful name of the lemma comes from imagining the graph of the function g as a mountainous landscape, with the sun shining horizontally from the right. The set E consist of points that are in the shadow.

Proof[edit]

The set E is open, so it is composed of a countable disjoint union of intervals (an, bn).

The main step is to show that g(bn) ≥ g(x) for x in (an, bn). If not take x with g(bn) < g(x). Let A be the closed subset of [x,bn] consisting of points y such that g(y) ≥ g(x). It contains x but not bn. It has a largest element, z say. Since z lies in E, there is a y with z < y < b and g(y) > g(z). Since bnE, g(t) ≤ g(bn) if bntb. Since g(y) > g(z) ≥ g(x) > g(bn), y must lie in (z, bn). That contradicts the maximality of z. Hence g(bn) ≥ g(an).

If ana, the reverse inequality holds. In fact since anE, g(y) ≤ g(an) if anyb. So g(bn) ≤ g(an). Hence g(bn) = g(an). If g(x) = g(an) at an interior point, then g(y) ≤ g(x) for x < y < b, contradicting xE.

Notes[edit]

  1. ^ Stein 1998
  2. ^ See:

References[edit]