Let $w_f$ be a distribution function of $f$ on $E\in\Bbb{R^n}$,$$w_f(\alpha) := \mu\left(\{\mathbf{x}\in E ~|~ f(\mathbf{x} \gt \alpha\}\right)$$
From triangle inequality, $$|f| \le |f-f_k|+|f_k|.$$
We can get the following inequality from the above; $$w_{|f|} \le w_{|f-f_k|}+w_{|f_k|}.$$
By drawing some examples on $\Bbb{R^2}$ domain, I understood, for $f \le g$, $$w_f \le w_g.$$
So, for $|f| \le |f-f_k|+|f_k|$, I can get $$w_{|f|} \le w_{|f-f_k|+|f_k|}.$$
However, I cannot proceed more. Please give me some ways.