There exists a sequence of Riemann integrable functions on [0, 1] whose pointwise limit is not Riemann integrable

Question

There exists a sequence of Riemann integrable functions on $[0, 1]$ whose pointwise limit is not Riemann integrable.

I think I need to construct some sequences but I don't know where to start.

Answer 1

Yes, there is such sequence. Let $\phi:\mathbb{Q}\to\mathbb{N} $ be a bijection from $\mathbb{Q}$ to $\mathbb{N}$

We define $$\left\lbrace \begin{array} .f_n(x) = 1 &\text{if} & x\in \mathbb{Q} \text{ and } \phi(x)<n\\ f_n(x) = 0 & & \text{elsewhere} \end{array} \right.$$

And you have

$$f_n \to \mathbb{1}_\mathbb{Q}$$ that is not Riemann integra

Answer 2

Here is a theorem related, a bounded function on $[a,b]$ is Riemann integrable iff it is continuous $\lambda$-a.e., where $\lambda$ denotes the Lebesgue measure.