# 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. – James Feb 16 at 10:41
• Yes, but what are the specific examples? – Eunice Feb 16 at 10:50
• I will let you know if I can construct an example. – James Feb 16 at 11:33
• Okay, thank you. :) – Eunice Feb 16 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, $$\liminf_{n\to+\infty} \int_0^3 f_n(x)\,dx = 1 \qquad \text{and}\qquad \limsup_{n\to+\infty}\int_0^3 f_n(x)\,dx = 2.$$