Q1. Consider $f\in L^2([0,1], R)$ with $ ||f||^2=\int f(x)^2d\mu(x)$
Consdier a subclass of simple functions $f= \sum_{i=1}^n a_i \chi_{A_i}$ where $A_i\in \Sigma$ (on $[0,1]$) and $A_n\subseteq...\subseteq A_2\subseteq A_1$,
Is the subclass dense in $L^2([0,1], R)$?
Q2. Consider $f\in L^2([0,1], R^2)$ with $ ||f||^2=\int \langle f(x),f(x) \rangle^2 d\mu(x)$
If the class of simple functions $f=(f_1,f_2)=( \sum_{i=1}^m a_i \chi_{A_i}, \sum_{j=1}^n b_j \chi_{B_j})$ is dense, i.e. for any $\epsilon>0$, there is $f\in L^2([0,1], R^2)$ such that $||f-f^*||<\epsilon$, for all $f^*\in L^2([0,1], R^2)$.
Could I further make such $f$ satisfying $m=n$ and $a_i=b_i$ for all $i$?
Thanks.