Let $(\mathbb{X},\mathcal{M},\mu)$ be a $\sigma$-finite measure space. Let $f_n \in L^{1}(\mu)$ be non-negative functions satisfying $$\int_{\mathbb{X}} f_{n}d\mu=1$$
(a) Show that it is not necessarily true that $f_{n}/n \to 0$ a.e.
(b) Show that $f_{n}/n^{2} \to 0$ a.e.
I solved (b) using the fact that $\sum_{n=1}^{\infty} \frac{1}{n^2}<\infty$ .
So naturally in order to solve (a) I tried to find some example that leads to the fact that $\sum_{n=1}^{\infty} \frac{1}{n}=\infty$ . But any example I've tried failed (for instance sequences of triangles with the area of 1).
Thanks!
How about this example:
Let $\mathbb{X}=[0,\infty)$ and $m$ Lebesgue measure.
Let $a_n=1/n$ and $n_1=1$.
Since $\sum_{n=1}^{\infty} \frac{1}{n}=\infty$ we can find a subsequence $\{a_{n_k}\}$ such that $S_k<S_{k+1}$ for every $k$ and $S_k \to \infty$, where $S_1=a_{n_1}=1$ and $S_{k}=\sum_{n=n_{k-1}+1}^{n_k} a_{n}$ for every $k>1$.
Then we construct a sequence of functions $\{g_n\}$ as follows: $$\chi_{[0,a_{n_1}]}, \chi_{[0,a_{n_1+1}]},\chi_{[a_{n_1+1},a_{n_1+1}+a_{n_1+2}]},..., \chi_{[\sum_{j=n_1+1}^{n_2-1}a_j,\sum_{j=n_1+1}^{n_2}a_j]}, \chi_{[0,a_{n_2+1}]},\chi_{[a_{n_2+1},a_{n_2+1}+a_{n_2+2}]}, ... $$
So $\int_{\mathbb{X}} g_{n}dm=1/n$ for every $n$, but since $S_k<S_{k+1}$ and $S_k \to \infty$ we get that $g_n(x)$ does not converge for any $x \in \mathbb{X}$, since there are infinitely many $n$ for which $g_n(x)=1$ and infinitely many for which $g_n(x)=0$.
So $f_n=n \cdot g_n$ should work.
Is this a good solution? Thanks!