Rising sun lemma

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:
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

We need a lemma: Suppose at some point z < d.
Since z ∈ S, there is a y in < g.
If y ≤ d, then g would not reach its maximum on at z.
Thus, y ∈ ≤ g < g.
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.
It follows immediately from the lemma that g < g for x in
.
Since g is continuous, we must also have g ≤ g.
If a_k ≠ a or a ∉ S, then a_k ∉ S,
so g ≥ g, for otherwise a_k ∈ S.
Thus, g = g in these cases.
Finally, if a_k = a ∈ S, the lemma tells us that g < g.