Rising Sun Lemma

The rising sun lemma, due to Frigyes Riesz, is a lemma of mathematical analysis used in the proof of the Hardy-Littlewood maximal theorem and serving as a one-dimensional precursor to the Calderon-Zygmund lemma. For a real-valued continuous function g on an interval [a, b], the lemma identifies the open set of points that lie, so to speak, in shadow when the graph of g is imagined as a landscape lit by a sun shining horizontally from the right, and shows that this set decomposes into a countable union of disjoint intervals with matching endpoint values of g. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/