Help with the last step in proving Hardy–Littlewood maximal inequality

Question

I'm following the proof of the Hardy–Littlewood maximal inequality on Wikipedia and I'm having trouble with the last step. It says that

$$2C_p\int_{t = 0}^{\infty}{\int_{|f| > t/2}{t^{p - 2}|f|dx}dt} = C_p \| f \|_p^p$$

I am not seeing how to justify this. I feel like there is something simple here but I'm not seeing it.

Answer 1

This is a VERY back-of-the-envelope calculation. But I think the general idea is to use Fubini theorem to flip the order of integration: