Let $(E,\mathcal{A},\mu)$ be a measure space such that $\mu(E)<+\infty$.
Is it possible to find an example of a sequence $f_n$ converging uniformly to $f$ (with no assumption if they are non-negative, integrable) such that $\lim_n\int_Ef_nd\mu \neq \int_Efd\mu$? (maybe for $n, f_n,$ is non-negative and the integral can be $+\infty,$ for other $n,f_n$ takes values in $\mathbb{R}$ and is not integrable so we can't define the integral)
Any suggestions are appreciated.
No. If $f_n$ converges uniformly to $f$ on $E$, then $$ \left| \int_E f_n \,d\mu - \int_E f \,d\mu \right| \leq \int_E |f_n - f| \,d\mu \leq \mu(E) \cdot \sup_{x \in E} |f_n(x) - f(x)| $$ By definition of uniform convergence, the above sup tends to $0$ as $n \to \infty$.