Give an example of a sequence of measurable functions $\{f_n\}$ defined on a measurable set $E \subseteq \mathbb{R}^n$ such that the following strict inequalities hold: \begin{align} \int_E \liminf f_n < \liminf \int_E f_n < \limsup \int_E f_n < \int \limsup f_n. \end{align}
MY ATTEMPT : Take $f_n = \chi_{[n, n+1]}$, then $\liminf f_n = 0$ and $\int_{\mathbb{R}} f_n= 1$, so we have $\int_{\mathbb{R}} \liminf f_n = 0 < 1 = \liminf \int_{\mathbb{R}} f_n$. But $\liminf \int_{\mathbb{R}} f_n = \limsup \int_{\mathbb{R}} f_n$ still holds.
Can anyone give me some examples please?