Constructive Decomposition of Any L^1(a,b) Function as Sum of a Strongly Convergent Series of Integrable Functions Each One Positive or Negative Exactly in Open Sets

Abstract

Researchers dealing with real functions f (·) ∈ L1 (a, b) are often challenged with technical difficulties on trying to prove statements involving the positive f+ (·) and negative f− (·) parts of these functions. Indeed, the set of points where f (·) is positive (resp. negative) is just Lebesgue measurable, and in general these two sets may both have positive measure inside each nonempty open subinterval of (a, b). To remedy this situation, we regularize these sets through open sets. More precisely, for each zero-average f (·) ∈ L1 (a, b), we construct, explicitly, a series of functions fi (·) having sum f (·) — a.e. and in L1 (a, b) — in such a way that, for each i ∈ {0, 1, 2, . . . }, there exist two disjoint open sets where fi (·) ≥ 0 a.e. and fi (·) ≤ 0 a.e., respectively, while fi (·) = 0 a.e. elsewhere. Moreover, its primitive \int^{t}f (·) becomes the sum of a strongly convergent series of nice AC functions. Applications to calculus of variations & optimal control appear in our next papers.