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.

[1] The lemma is stated as follows:[2] 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.

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).

Finally, if ak = a ∈ S, the lemma tells us that g(a) < g(bk).