A class of functions dense in $L^2$

Question

Suppose $f\in L^2([0,1],\Sigma,\mu)$. Is the class of all $$f=\sum_{i=1}^n \alpha_i (\chi_{A_i}-\chi_{[0,1]/A_i} )$$$A_i\in \Sigma$ to be dense in $L^2([0,1],\Sigma,\mu)$? Thanks.

Answer 1

Suppose $g\perp (\chi_{[0,x)}-\chi_{[x,1]})$ for all $x\in [0,1].$ Then $\int_0^x g = \int_x^1g, x \in [0,1].$ By the Lebesgue differentiation theorem, $g(x)=-g(x)$ for a.e. $x.$ Thus $g=0$ a.e. This shows that the subspace you defined is dense in $L^2.$