I am trying to show that if I can approximate two characteristic functions $\chi_A,\chi_B$ by simple functions involving only a particular set of characteristic functions, then I can approximate $\chi_A\chi_B$. Along the way I have run into the problem of trying to bound the $L^p$ norm of a product.
I am trying to show that if $D$ is a countable collection of sets, and $\Sigma$ is the $\sigma$-algebra generated by $D$, and $A$ is the algebra generated by $D$, then the closure of the set of simple functions involving only characteristic functions in $A$ covers simple functions involving only characteristic functions in $\Sigma$. It suffices to show the closure covers characteristic functions of sets in $\Sigma$, which is where this problem arises. –
We work in $(X,\Sigma,\mu)$ a finite measure space. Suppose $s$ can be chosen so that $\left\Vert \chi_A-s \right\Vert_p < \epsilon$ and$t$ can be chosen so that $\left\Vert\chi_B - t\right\Vert<\epsilon$, where $s$ and $t$ are simple. Is there some way that I can bound $\left\Vert (\chi_A-s)(\chi_B-t) \right\Vert_p$ in a way that only involves $\epsilon$ and possibly $\mu(X)$? I was able to show it can be bounded by $\epsilon^p \max\{(\chi_A-s)(\chi_B-t) \}$ but this is not helpful because the goal is to take the limit as $\epsilon \to 0$ and the max could blow up because $s,t$ depend on $\epsilon$.