Let $A_n = [1-n,n-1]$. Define $$\chi_{\mathbb{R} \setminus A_n} := \begin{cases} 1 \qquad x \in \mathbb{R} \setminus A_n \\ 0 \qquad \text{ else } \end{cases}$$
Is it true that $$ \lim_{n \to \infty} \int_{\mathbb{R}} \chi_{\mathbb{R} \setminus A_n} \, d \mu = \infty$$ yet $$\int_{\mathbb{R}} \lim_{n \to \infty} \chi_{\mathbb{R} \setminus A_n} = 0?$$
In both cases, this is a Lebesgue Integral. Think of this function as the line $y=1$ 'falling' to $0$.