Give an example of a sequence of non-negative Lebesgue measurable functions $f_n:[0,1] \rightarrow \mathbb{R}$ in the form $f_n=c_n \chi(F_n)$, where $F_n \subseteq [0,1]$ such that $||f_n||_2 \rightarrow \infty$ and $||\sqrt{f_n}||_2 \leq 1$. Justify your example. Here $||h||_2$ stands for the $L^2$-norm of function h with respect to the Lebesgue measure on $[0,1]$.
So I have been trying to do this question that seems relatively straight forward, but I have not been able to find the function that satisfies both of these conditions.
Here is the working that I have so far (please let me know if this is correct!) \begin{equation} ||f_n||_2=(\int|f_n|^2 d\lambda )^{1/2} = ( \int |c_n \chi(F_n) |^2 d\lambda)^{1/2}= (\int|c_n|^2 \chi(F_n) d\lambda)^{1/2} \end{equation}
and
\begin{equation} ||\sqrt{f_n}||_2=(\int|\sqrt{f_n}|^2 d\lambda )^{1/2} = ( \int |\sqrt{c_n \chi(F_n)} |^2 d\lambda)^{1/2}= (\int|c_n| \chi(F_n) d\lambda)^{1/2} \end{equation}
It is also relevant that $\int \chi(F_n)d\lambda =\lambda(F_n). $
Help! I feel as though I am close to the solution or making a really obvious mistake!
Let $p_n$ denote the Lebesgue measure of $F_n$. We assume that $c_n$ is non-negative. We can put the constant out of the integral and we get that $$\left\lVert f_n\right\rVert_2=c_np_n^{1/ 2}\mbox{ and }\left\lVert \sqrt{ f_n} \right\rVert_2= c_n^{1/2} p_n^{1/ 2}.$$ If we take $p_n=1/c_n$, then the second constraint holds. For the first one, we have $$\left\lVert f_n\right\rVert_2=c_np_n^{1/ 2}=c_n c_n^{-1/2}=c_n^{1/2}.$$ The choice $$c_n=n^2\mbox{ and }F_n=\left[0,\frac 1{n^2}\right] $$ gives what we want.