The weak controlled convergence theorem for Henstock integrals (Q1825295)

If a sequence of functions \(\{f_ n\}\) satisfies the following conditions: \((i)\quad f_ n(x)\to f(x)\) a.e. in [a,b], where all of \(f_ n\) are Henstock integrable on [a,b], (ii) the primitives \(F_ n\) of \(f_ n\) are \(ACG_*\) uniformly in n, (v) the primitives \(F_ n(x)\) converge on a dense subset of [a,b] to a continuous function F(x) on [a,b], then \(F_ n(x)\) converge to F(x) nearly everywhere (i.e. everywhere except for a countable subset) in [a,b] and f is Henstock integrable on [a,b] with primitive F. If a sequence \(\{f_ n\}\) satisfies conditions (i), (ii) and (vi) \(F_ n(x)\) converges to F(x) on a dense subset of [a,b], then \(F_ n(x)\to F(x)\) nearly everywhere in [a,b] and the limit function F is continuous nearly everywhere in [a,b]. If, in addition, the series \[ \sum_{x\in (a,b)}| F(x+)-F(x-)| +| F(a+)-F(a)| +| F(b)-F(b- )| \] is finite, then f is Henstock integrable on [a,b] with primitive \(F(x-)-\sum_{y<x}(F(y+)-F(y-)).\)