Show that there are $f \in L^1(\Bbb{R}^d,m)$ and a sequence $\{f_n\}$ with $f_n \in L^1(\Bbb{R}^d,m)$ such that $\|f_n - f\|_{L^1} \to 0$, but $f_n(x) \to f(x)$ for no $x$.
Asked
Active
Viewed 301 times
1
-
@Sara Tancredi The edit you suggested was actually an answer. This question itself is a duplicate, but you may like to post your answer to the linked question instead. – Tom Oldfield May 06 '13 at 23:52
1 Answers
2
Let us see the idea for $\mathbb{R}$ . Let $f_1$ be a characteristic function of $[0,0.5]$, $f_2$ --- of [0.5,1], $f_3$ of [0,0.25] and so on. They form the desired sequence on $L^1([0,1])$. The rest is an exercise because $\mathbb{R}$ can be covered by a countable number of intervals.
Halil Duru
- 1,347
Przemysław Scherwentke
- 13,668
- 5
- 35
- 56