# How to bound $L^p$ norm of a product

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 $\epsilon$.

-

Let $s_n$ simple such that $\lVert\chi_A-s_n\rVert_p\leq n^{—1}$, and $t_n$ such that $\lVert \chi_B-s_n\rVert_p\leq n^{—1}$. Then $\lVert s_n\rVert_p\leq n^{—1}+\mu(A)\leq 1+\mu(X)$ and similarly $\lVert t_n\rVert_p\leq 1+\mu(X)$. So $$\lVert \chi_A\chi_B-s_nt_n)\rVert_p\leq \lVert \chi_A(\chi_B-t_n))\rVert_p+\lVert t_n(\chi_A-s_n)\rVert_p\leq \lVert \chi_B-t_n\rVert_p+\lVert t_n\rVert_{p'}\lVert \chi_A-s_n\rVert_p,$$ where $\frac 1{p'}+\frac 1p=1$. So if $p<p'$ and $\mu(X)=1$, we are done. (the assumption $\mu(A)$ can be done)
By duality argument you mean to use the fact that $\left\Vert t_n \right\Vert_{p'} = \sup_{\left\Vert g \right\Vert_p = 1} \int_X t g d\mu$? I don't see how to apply this in the case where $p > p'$. – nullUser Oct 25 '12 at 14:57
Also how did you get $\left\Vert t_n(\chi_A-s_n)\right\Vert_p \leq \left\Vert t_n \right\Vert_{p'}\cdot \left\Vert \chi_A - s_n \right\Vert_p$? I don't see how this follows from Holder. – nullUser Oct 25 '12 at 15:05
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. – nullUser Oct 25 '12 at 15:19
And you want an approximation in $L^p$, I guess? To see that, you can use the lemma: if $(X,\mathcal B,\mu)$ is a finite measure space and $\mathcal A$ an algebra generating $\mathcal B$, then for each $\varepsilon>0$ and $B\in \mathcal B$, we can find $A\in\mathcal A$ such that $\mu(A\Delta B)\leq \varepsilon$. – Davide Giraudo Oct 25 '12 at 15:28