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.
The lemma is stated as follows:[1]
Suppose g is a real-valued continuous function on the interval [''a'',''b''] and S is the set of x in [''a'',''b''] such that there exists a y∈(x,b] with g(y) > g(x). (Note that b cannot be in S, though a may be.) Define E = S ∩ (a,b).
Then E is an open set, and it may be written as a countable union of disjoint intervals
E=cupk(ak,bk)
such that g(ak) = g(bk), unless ak = a ∈ S for some k, in which case g(a) < g(bk) for that one k. Furthermore, if x ∈ (ak,bk), then g(x) < g(bk).
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.
We need a lemma: Suppose [''c'',''d'') ⊂ ''S'', but d ∉ ''S''. Then ''g''(''c'') < ''g''(''d''). To prove this, suppose ''g''(''c'') ≥ ''g''(''d''). Then ''g'' achieves its maximum on [''c'',''d''] at some point z < d.Since z ∈ S, there is a y in (z,b] with g(z) < g(y). If y ≤ d, then g would not reach its maximum on [''c'',''d''] at z.Thus, y ∈ (d,b], and g(d) ≤ g(z) < g(y).This means that d ∈ S, which is a contradiction, thus establishing the lemma.
The set E is open, so it is composed of a countable union of disjoint intervals (ak,bk).
It follows immediately from the lemma that g(x) < g(bk) for x in (ak,bk).Since g is continuous, we must also have g(ak) ≤ g(bk).
If ak ≠ a or a ∉ S, then ak ∉ S, so g(ak) ≥ g(bk), for otherwise ak ∈ S. Thus, g(ak) = g(bk) in these cases.
Finally, if ak = a ∈ S, the lemma tells us that g(a) < g(bk).