Let $ f \in L^1\left(\mathbb{R}^n\right) $. Prove that for any $ \lambda > 0 $,$$\left|\left\{x \in \mathbb{R}^n : M f(x) > \lambda \right\}\right| \leq \frac{C}{\lambda} \int_{(|f| > \lambda / 2)} |f(x)| \, dx$$where the constant $C$ is independent of $\lambda$ and $f$.
I have thought of two ways to try to solve this problem but neither has been successful.
First, I tried to use weak-(1,1)-inequality. Trying to find a function $g$ such that $$\left|\left\{x \in \mathbb{R}^n : M f(x) > \lambda \right\}\right| \leq|\{x \in \mathbb{R}^n : Mg(x) > \lambda\}| \leq \frac{C}{\lambda} \int_{\mathbb{R}^n} |g(x)| \, dx=\frac{C}{\lambda} \int_{(|f| > \lambda / 2)} |f(x)| \, dx$$And a (wrong) example is $g(x) = f(x) \chi_{\{|f(x)| > \lambda/2\}}$. It cannot solve this problem.
Second, I tried to use Vitali Covering Lemma. We have $$\left|\left\{x \in \mathbb{R}^n : M f(x) > \lambda \right\}\right|\subset \bigcup_{i} B(x_i, 5r_i)$$Then $$\left|\left\{x \in \mathbb{R}^n : M f(x) > \lambda \right\}\right|\leq\frac{5^n}{\lambda} \sum_i \int_{B(x_i, r_i)} |f(y)| \, dy$$But I cannot relate it to $\{|f| > \lambda / 2\}$.
How can I solve this? Or both of the two ways are wrong? Or someone can find the source of this problem?