Is it possible to interchange sum and integral when the series doesn't converge uniformly?

Question

Suppose a series $\sum f_n$ of integrable functions on $[0,1]$ does not converge uniformly, is it possible that $$\sum_{n=1}^\infty \int_0^1 f_n(x)dx = \int_0^1\sum_{n=1}^\infty f_n(x)dx$$ still holds?

Answer 1

Let's consider $$f(x)=\sum\limits_{n=1}^{\infty}\left(\frac{nx}{1+n^2x^2} - \frac{(n-1)x}{1+(n-1)^2x^2}\right)$$ this series have continuous sum $0$, on interval $[0,1]$, though converges non uniformly and holds $$\lim\limits_{n \to \infty}\int\limits_{0}^{1}\frac{nx}{1+n^2x^2} \,dx=\lim\limits_{n \to \infty}\frac{\ln (1+n^2)}{2n}=0=\int\limits_{0}^{1}f(x)\,dx$$