Space of Simple P-Integrable Functions is Everywhere Dense in Lebesgue Space

Theorem
Let $\struct {X, \Sigma, \mu}$ be a measure space.

Let $p \in \R$, $p \ge 1$.

Let $\map {\LL^p} \mu$ be Lebesgue $p$-space for $\mu$.

Let $\map \EE \Sigma \cap \map {\LL^p} \mu$ be the space of $\Sigma$-simple, $p$-integrable functions.

Then $\map \EE \Sigma \cap \map {\LL^p} \mu$ is everywhere dense in $\map {\LL^p} \mu$.

Proof
For $n \in \N$, we define $D_n : \R_{\ge 0} \to \R_{\ge 0}$ by:
 * $\map {\Delta _n} y := \begin{cases}

0 & : y \ge 2^n \\ \dfrac k {2^n} & : y \in \hointr {\dfrac k {2^n} } {\dfrac {k+1} {2^n} }, k = 0, 1, \ldots, 2^{2n}-1 \end{cases}$

Then, let:
 * $\map {f_n} x := \map \sgn {\map f x} \map \Delta {\size {\map f x} }$

where:
 * $\map \sgn \cdot$ denotes the signum function
 * $\size \cdot$ denotes the absolute value.

Then, for all $x \in X$:
 * $\ds \lim_{n \mathop \to \infty} \map {f_n} x = \map f x$

and:
 * $\size {\map {f_n} x} \le \size {\map f x}$

Moreover:
 * $f_n \in \map \EE \Sigma \cap \map {\LL^p} \mu$

In view of Riesz's Convergence Theorem, it suffices to show:
 * $\ds \lim_{n \mathop \to \infty} \norm {f_n}_p = \norm f_p$

It follows from Lebesgue's Dominated Convergence Theorem.