I have a question I would like to start a bounty on. But I would like to edit the question to add the possibility for a stronger result. The question I want to start a bounty on is this question:
For an integrable function, does there exist a sequence of partitions with this property.
But I would like to modify it like this:
For a partition $\Pi$ we define $f_{\Pi}(t)=\sum\limits_{i=0}^{n-1}f(t_i)\cdot1_{\{x: |f(x)|<\infty\}}(t_i)1_{x \in [t_i,t_i+1)}(t)$. I also want to know if there exists a sequence of partitions $\Pi_n$ such that $\int_{[a,b]}|f(t)-f_{\Pi_n}(t)|dt\rightarrow0$ or $\int_{[a,b]}|f(t)-f_{\Pi_n}(t)|^2dt\rightarrow0$.
Is it most appropriate to edit the question and include the new info, or ask a new quesiton which include all the info, but it will also include the former question? I am unsure if it is allowed to edit the question because it changes the question by a lot.
