Weak $L_1$ norm is different from $L_1$ norm on a probability space

Question

Can any one give an example of a probability space $(X , \mu )$ and functions $f_1, ...,f_n : X \to \mathbb R$ such that $ \| f_i \|_{ L^{ 1, \infty }} : = \sup_{ t > 0} t \lambda_{f_i} (t) \le 1$ for all $ 1 \le i \le n$ ( $\lambda_{f_i}(t)$ is the distribution function), but $ \|\sum_{ i =1}^ n f_i \|_{L^{1, \infty}} \ge cn\log n $ for some $ c > 0$.

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This tells me that weak $L_1$ norm is different from $L_1$ norm on a probability space in general. However, I could not find a example. Any help is appreciated.

Answer 1

Let $X = [0,1]$ with Lebesgue measure. Let $ f_k(t) = n/((k+j \bmod n)+1)$ if $t \in [(j-1)/n,j/n)$, where $a \bmod n$ is the remainder after dividing $a$ by $n$.