If $B$ is a finite ball in $\mathbb{R^n}$ and $f \in L^1(B)$, why do we have the following equality:
$$\int_1^\infty \frac{1}{t}(\int_{\{|f| > \frac{t}{2}\}}|f(x)|dx) dt = \int_B ( \int_1^{\text{max}\{2|f(x)|, 1\}} \frac{1}{t}dt)|f(x)|dx$$
This equality is showing up in the proof of the statement:
The maximal function of $f$ is in $L^1(B)$ iff $|f|\log(e + |f|) \in L^1(B)$
But I don't understand the above equality. I would appreciate any help in understanding it!