# Does $f_n \to f$ pointwisely imply $\int f_n \to \int f$ for conditionally Riemann integrable functions?

Let $$f_n:\mathbb{R}^n\to\mathbb{R}$$ be a sequence of conditionally (improper) Riemann integrable functions that pointwisely converges to $$f:\mathbb{R}^n\to\mathbb{R}$$ which is also conditionally Riemann integrable. Does $$\int f_n \to \int f$$? If they were absolutely Riemann integrable or Lebesgue integrable, it would be true by the dominated convergence theorem. I'd like to know it is also true (with some additional conditions) for conditionally converging integrals.

Take $$f_n = n1_{\left(0,\frac{1}{n}\right]}$$ then $$f_n \to f$$ pointwise with $$f \equiv 0$$
But $$\int f_n = 1 \not\to 0 = \int f$$
• How about $0<\int |f| <\infty$? Can this be a sufficient condition? Intuitively, this may prohibit accumulation of the mass of $f_n$ in an infinitely small region. – Balbadak Nov 14 '18 at 6:55
• OFC not… take $$\tilde{f_n} = 1_{[0,1]} + f_n$$ then $$\tilde{f} = 1_{[0,1]}$$ and so your condition holds but still $$\int \tilde{f_n} = 2 \not\to 1 = \int \tilde{f}$$ – Gono Nov 14 '18 at 11:50