A refinement of the controlled convergence theorem for Henstock integrals (Q1100298)

Let \(f,f_ 1,f_ 2,..\). be real functions defined on [a,b]. Consider the following conditions: \((i)\quad f_ n(x)\to f(x)\) a.e. in [a,b], (ii) the primitives \(F_ n\) of \(f_ n\) are \(ACG_*\) uniformly in n, (iii) the primitives \(F_ n\) converge uniformly on [a,b], (iv) the primitives \(F_ n(x)\) converge pointwise at each x to a continuous function F(x) in [a,b]. \textit{P. Y. Lee} and \textit{T. S. Chew} [Bull. Lond. Math. Soc. 17, 557-564 (1985; Zbl 0553.26002)] showed that under conditions (i)-(iii) f is Henstock integrable on [a,b] and \[ \int^{b}_{a}f_ n(x)dx\to \int^{b}_{a}f(x)dx. \] The same is proved under conditions (i), (ii) and (iv) in this paper.