Let $f \in L^1+L^{\infty}$. We are given that $\|f\|_{L^1+L^{\infty}}=\inf\{\|g\|_1+\|h\|_{\infty} : g\in L^1, \ h \in L^{\infty} \ \text{and} \ f=g+h \}$ and $$\|f\|_+=\dfrac1t+\int\limits_{\{|f|>\frac1t\}}\left(|f|-\dfrac1t\right)d\mu \quad \text{where} \ \dfrac1t=\inf\left\{\frac1t : \mu\{|f|\geq\frac1t\}\geq 1\right\} \text{are two norms on} \ L^1+L^{\infty} $$
I am trying to show that $\|f\|_A=\inf_{k>0}\left\{ \dfrac1k+\int\limits_{\{|f|>\frac1k\}}\left(|f|-\dfrac1k\right)d\mu\right\}=\|f\|_+$ or $\|f\|_A=|f\|_{L^1+L^{\infty}}$.
I am in showing $\|f\|_A=\|f\|_+$ is easier. $\|f\|_A\leq\|f\|_+$ by the definition of infimum if I am not wrong. How can I prove converse of this inequality?
If I am totally wrong, could you please lead me the way for showing other equality?
I am grateful for any help