Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $\epsilon$.

share|improve this question

1 Answer 1

up vote 1 down vote accepted

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)

If it's not the case, then use a duality argument.

This show that the collection of approximable sets is stable by finite intersection provided so are the approximating sets.

share|improve this answer
    
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
1  
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
    
Yes, you are right. Before I try to salvage my attempt, I have a question: what do you know about the approximating simple functions? Are you in a particular context? –  Davide Giraudo Oct 25 '12 at 15:13
1  
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
1  
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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.