Hardy-Littlewood maximal theorem (Marcinkiewicz) I have two pages from a book called "Garnett" and I will present Hardy-Littlewood maximal theorem in class on Wednessday.
The theorem is stated:
if $f\in L^p(\mathbb{R}), 1 \leq p \leq \infty,$ then $Mf(t)$is finite a.e.
b) if $f\in L^p(\mathbb{R}), 1 < p \leq \infty,$ then $Mf\in L^p(\mathbb{R})$ and
$$ \|Mf\|_p \leq A_p \|f\|_p,$$
where $A_p$ depends only on $p$. 
Here $Mf(t)$ is Hardy-Littlewood maximal funcion.
What I dont understand is on the next page in the proof of Marcinkiewicz.
First they say we bound $\nu(E_\lambda)$ by $\nu(B_\lambda)$ +  $\nu(C_\lambda)$
and use Lemma 4.1. I don't know this lemma and it is not in my copied paper, can someone please explain.
Second in the case of $p_1 =\infty$ 
$$\|Tf\|_p^p = \int_0^\infty p \lambda^{p-1} \nu(E_\lambda) d\lambda$$
here $E_\lambda = \{y: |Tf(y) |> \lambda \}.$
This seems like some classical trick of variable change. I think I seen similar things before, can someone explain of prove that this "obvious" thing is holds true. Thanks for the help

I know this post is old, but here is the solution if you were still wondering. The idea is to use Fubini's theorem. Let $\chi_A$ be the characteristic function of $A \subset \mathbb{R}$.
$||Tf||_p^p = \int_{\mathbb{R}}|Tf(x)|^p dx = \int_\mathbb{R} \int_0^{Tf} p\lambda^{p-1} d\lambda dx = \int_\mathbb{R} \int_0^\infty p\lambda^{p-1}\chi_{E_\lambda} d\lambda dx = \int_0^\infty p\lambda^{p-1} d\lambda \int_\mathbb{R} \chi_{E_\lambda}dx $ 
The claim is immediate from there.
