Stein & Shakarchi's Real analysis Hardy-Littlewood Maximal function$ f^{*}(x) \geq |f(x)|$ a.e.

In Stein & Shakarchi's Real analysis book, I failed to see that

Note that as an immediate consequence of the theorem applied to $|f|$, we see that $f^*(x) \geq |f(x)|$ for a.e. x, with $f^*$ the maximal > function.

on page 105.

The theorem it refers to is the Lebesgue differentiation theorem If $f \in L^1$, then $ \lim\limits_{m(B)\rightarrow 0, x \in B} \frac{1}{m(B)}\int_{B}f(y)dy = f(x), \mbox{ for a.e. x} $

Why is it immediate?

Recall that the maximal function is defined by $$f^*(x)=\sup_{x\in B}\frac{1}{m(B)}\int_B|f(y)|\,dy,$$ where the supremum is taken over all balls containing $x$.

Applying the Lebesgue differentiation theorem to $|f|$, we get $$\lim_{m(B)\to 0, x\in B}\frac{1}{m(B)}\int_B|f(y)|\,dy=|f(x)|$$ for almost every $x$.

Thus, since $f^*$ is the supremum of the quantity $\frac{1}{m(B)}\int_B|f(y)|\,dy$ that shows up in the limit, it must be bounded below by $|f(x)|$ almost everywhere.