# give an example such that the strict inequalities $\int_E \liminf f_n < \liminf \int_E f_n < \limsup \int_E f_n < \int \limsup f_n$ hold

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?

• That essentially means we need to find a sequence of function that does not converge. Also, we need to make sure that the sequence of the integral also does not converge. This is the idea. Feb 16, 2020 at 10:41
• Yes, but what are the specific examples? Feb 16, 2020 at 10:50
• I will let you know if I can construct an example. Feb 16, 2020 at 11:33
• Okay, thank you. :) Feb 16, 2020 at 12:52

Let $$E=[0,3]$$ and consider $$f_n \begin{cases} \chi_{[0,2]} & \text{ if } n \text{ is even},\\ \chi_{[2,3]} &\text{ otherwise}. \end{cases}$$ Then, $$\liminf_{n\to +\infty} f_n = 0 \qquad\text{and}\qquad \limsup_{n\to +\infty} f_n = 1,$$ which implies $$\int_0^3 \liminf_{n\to +\infty} f_n = 0 \qquad \text{and}\qquad \int_0^3\limsup_{n\to +\infty} f_n = 3.$$ Moreover, $$\int_0^3 f_n(x)\,dx = \begin{cases} 2 & \text{ if } n \text{ is even},\\ 1 &\text{ otherwise}. \end{cases}$$ Hence, $$\liminf_{n\to+\infty} \int_0^3 f_n(x)\,dx = 1 \qquad \text{and}\qquad \limsup_{n\to+\infty}\int_0^3 f_n(x)\,dx = 2.$$