BSC4024108 Project Details
| Supervisor | Ng Wee Leng |
| Project Code | BSC4024108 |
| Title of Project | Primitives of Henstock-Kurzweil Integrable Functions |
| Description | The Henstock-Kurzweil integral is a generalization of the Riemann integral, and is more general than the Lebesgue integral. In particular, a function f is Lebesgue integrable if and only if f and its absolute value |f| are Henstock–-Kurzweil integrable. If a function is Henstock–-Kurzweil integrable on a closed bounded interval [a,b] on the real line, it is so on every subinterval [c,d] of [a,b] and we denote its integral on [c,d] as (HK)∫dcf(t)dt. Furthermore, we can define a function F on [a,b] given byF(x)=(HK)∫xaf(t)dtfor x∈[a,b]. We call the function F the \emph{primitive} of f on [a,b]. It is well known that the set P of all primitives of Henstock-Kurzweil integrable functions on [a,b] is a subset of the space C[a,b] of all continuous functions on [a,b]. We shall verify that P is a topological vector space and prove that it is a dense subset of C[a,b]. |
| Pre-requisites | • Real analysis, linear space |
| References | [1] Kurzweil, J. & Jarnik, J., A Convergence Theorem for Henstock-Kurzweil Integral and its Relation to Topology, Bull Acad Royal de Belgique, Classe de Sci, 7-12 (1997), 217-223. [2] Ng, W.L., & Lee, P.Y., An Alternative De nition of the Henstock-Kurzweil Integral Using Primitives, New Zealand Journal of Mathematics, 48 (2018), 121-128. |