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