Give a function in $L^p$ but not in weak $L^p$

Question

I come across this problem;

1. Define the space Weak $L^p$ in the unit ball $B^n_1(0)$ in $\mathbb{R}^n$ for $1 <p <\infty$.

2. Show an example of a function $f(x)$ such that $f \in L^{p,w}(B_1^n(0),dx)$, but not in the classical $L^p(B_1^n(0))$

My solution

1. Consider the measure space $(B_1^n(0), \mathcal{B}(B_1^n(0)),dx)$, and let $f$ be a measurable real-valued function on $B_1^n(0)$. Let the distribution function of $f$ is denoted for $t >0$ by $\lambda_f : \mathbb R \to \mathbb R$ where

$$\lambda_f(t) = \left|\{x \in B_1^n(0): |f(x)|>t \} \right|.$$

Then a function $f$ is said to be in the Weak $L^p(B_1^n(0))$ space if $\exists C>0 $ such that $\forall t >0$, then

$$\lambda_f (t) \leq \frac{C^p}{t^p}.$$

I could not find an example. I assume I can not imagine the behavior of the functions in weal Lebesgue space. Any help is appreciated.