I have been given the following questions:
Consider the following measure spaces (X, µ):
I. $X = [0, 1], µ$ is Lebesgue measure.
II. X = [0, ∞), µ is Lebesgue measure. Show examples of functions f1, f2, . . . and $f: X \rightarrow R$ such that $f_{n}$ converges to f
a.) in $L^1$ but not in $L^2$ , b.) in $L^2$ but not in $L^1$
now for the I part , I kind of have an idea, my solution is: a. $nχ_{[0,\frac{1}{n^2}]}$ , which is n between on the interval $[0,\frac{1}{n^2}]$ and 0 everywhere.
b. I said we cannot have such a scenario
now my question is for the II question, whats confusing me is the space that I have to work on, I have no starting point, any hint would be appreciated.
$f_{n}=\dfrac{1}{n}\chi_{[0,n]}$, $\|f_{n}\|_{L^{2}[0,\infty)}=\dfrac{1}{n}\rightarrow 0$ but $\|f_{n}\|_{L^{1}[0,\infty)}=1$.
For counting measure, consider \begin{align*} a_{1}&=(1/2,0,0,...)\\ a_{2}&=(1/2,1/3,0,...)\\ a_{3}&=(1/2,1/3,1/4,0,...) \end{align*} and $a=(1/2,1/3,1/4,1/5,...)$, then $\|a_{n}-a\|_{l^{2}}=\displaystyle\sum_{k\geq n+1}\dfrac{1}{k^{2}}\rightarrow 0$ but $\|a_{n}-a\|_{l^{1}}=\infty$.