I am not understanding a proof of a lemma in the following text Chung,Williams,2014, "Introduction to Stochastic Integration".
Settting
Consider $\mathscr{L}^2(\mathbb{R}^+\times\Omega,P,\mu_M)$ where $\mu_M((s,t)\times F)=E(\mathbb{1}_F(M_t-M_s)^2)$ and $\varepsilon$ is the set of stochastic integrable random variables $X\in \varepsilon$ such that $\int X dM=\sum_{i=1}^{n}c_j\mathbb{1}_{F_j}(M_{t_j}-M_{s_j})$.
Theorem 2.3. For $X\in\varepsilon$ we have the isometry:
$$E \left\{ \left(\int X dM\right)^2\right\}=\int X d\mu_M$$
Lemma 2.4. The set of $R$-simple processes $\varepsilon$ is dense in the Hilbert space $\mathscr{L}_2$.
Proof: Since $P$ is generated by the ring $\mathscr{A}$ and $\mu_M$ is $\sigma$-finite, then for each $\epsilon > 0$, and $A \in P$ such that $\mu_M(A) < \infty$, there is $A_1\in \mathscr{A}$ such that $\mu_M(A\triangle A_1)<\epsilon$ where $A\triangle A_1$ is the symmetric difference of A and A1 (see Halmos [37; p. 42, 49]). It follows that any $P$-simple function in $\mathscr{L}_2$ can be approximated arbitrarily closely in the $\mathscr{L}_2$-norm by functions in $\varepsilon$. The proof is completed by invoking the standard result that the set of'P-simple functions is dense in $\mathscr{L}_2$
In the proof of the above theorem I fail to see why "It follows that any $P$-simple function in $\mathscr{L}_2$ can be approximated arbitrarily closely in the $\mathscr{L}_2$-norm by functions in $\varepsilon$.". Since it follows as the conclusion of the symmetric difference, it is related, but I fail to see how.
Question:
Can someone explain me or help me complete this proof?
Thanks in advance!