$L^p$ space, simple function and density.

The measurable simple functions $$\mathcal{S}$$ are dense in $$L^p(X,\mathcal{A,\mu}).$$ The key result on which this is based is the following theorem.

Theorem. Let $$(X,\mathcal{A})$$ a measurable space, $$f\colon X\to \mathbb{\overline{R}}$$. Then exists a sequence $$\{s_n\}$$ of simple functions on $$X$$ such that $$\lim_{n\to\infty} s_n(x)=f(x)$$ in $$X$$. Moreover,

$$(i)\;$$ if $$f$$ is measurable, then $$s_n$$ is measurable for all $$n$$;

$$(ii)\;$$ if $$f$$ is nonegative, the sequence $$\{s_n\}$$ is increasing and we have $$0\le s_n\le f\quad n\in\mathbb{N}.$$

$$(iii)\;$$ if $$f$$ is bounded, then the convergence is uniform.

Now, in the proof it is supposed that, at first, that $$f$$ is bounded and nonnegative; then it is built $$s_n:=\sum_{k=0}^{2^n-1}\frac{k}{2^n}\chi_{E_k^n},$$ where $$E_k^n:=\bigg\{x\in X\;\bigg|\; \frac{k}{2^n}\le f(x)<\frac{k+1}{2^n}\bigg\}.$$

If $$f$$ it is not bounded(non negative), then it is built $$s_n=n\chi_{E_I^n}+\sum_{k=0}^{n2^n-1}\frac{k}{2^n}\chi_{E_k^n},$$ where $$E_I^n:=\{x\in X\;|\; f(x)\ge n\}.$$

Question. In light of the above, since the $$s_n$$ coefficients are always rational, can I conclude in the same way that the simple functions with rational coefficients (obviously with limited support) are dense in $$L^p(X)$$? If I can't finish this, how do I show it then?

• Yes you can conclude that.
– jvc
Commented May 20, 2020 at 15:36
• For the bounded nonnegative $f$ it appears you are assuming $f\le 1$ eveywhere. For your questions are you asking how to prove such simplie functions are dense in $L^p$? And I'm not sure what you mean with the last question.
– zhw.
Commented May 22, 2020 at 18:21
• @zhw In generel, I want to show that simple functions with rational coefficients are dense in $Lp$. Now, there is the above theorem, which under some hypothesis shows that a measurable function can be approximated through simple function, where this simple function are those in question. I observed that those simple functions have rational coefficient, and wondered if this was enough for the above purpose. Commented May 23, 2020 at 10:10

Yes, the theorem you cite leads to the result on density. Suppose $$1\le p <\infty$$ and $$f\in L^p.$$ Write $$f= f^+-f^-.$$ Then there exist simple nonnegative $$s_n,t_n$$ with rational coefficients such that $$0\le s_n \le f^+,$$ $$0\le t_n \le f^-$$ with $$s_n \to f^+,$$ $$t_n \to f^-$$ pointwise everywhere. Note that
$$|s_n-f^+|^p = (f^+ -s_n)^p \le (f^+)^p\,\,\text{everywhere}.$$
Thus by the DCT, $$\|s_n-f^+\|_p \to 0,$$ i.e., $$s_n\to f^+$$ in $$L^p.$$ Similarly, $$t_n\to f^-$$ in $$L^p.$$ It follows that $$s_n-t_n \to f^+ - f^-=f$$ in $$L^p.$$ Since each $$s_n+t_n$$ is a simple function with rational coefficients, we're done.
I'll leave the $$p=\infty$$ case to you for now. Ask if you have any questions.