Find an increasing sequence of step functions which converges pointwise everywhere to the function$$ f(x)=\chi_{[0,1] \cap \Bbb Q} (x) · x. $$

I know how to approximate a non-negative measurable function by an increasing sequence of non-negative simple functions. But I do not know the way of approximating any non-negative measurable function by an increasing sequence of step functions. Would anybody please help me finding this? Actually I want to know the geometrical approach behind that kind of approximations which will enable me to solve these types of problems on my own.

Please help me in this regard. Any geometrical approach will be appreciated.

Thank you very much.

  • $\begingroup$ I would suggest that you think about what a nonnegative step function which is less than $f$ can look like. Can you write down any interesting examples? $\endgroup$ – Eric Wofsey Apr 30 '18 at 4:35
  • $\begingroup$ Then $g$ is a step function for all $n \in \Bbb N$ such that $g(x) \leq f(x)$ for all $x \in \Bbb R$. Isn't it so @Eric Wofsey? $\endgroup$ – Dbchatto67 Apr 30 '18 at 5:36
  • $\begingroup$ No. $g(x)\not\leq f(x)$ if $x\in (1/n,1)$ is irrational. $\endgroup$ – Eric Wofsey Apr 30 '18 at 6:26
  • $\begingroup$ Oh! I see. I am trying to prove by taking degenerate step function as no restriction is given in the question. $\endgroup$ – Dbchatto67 Apr 30 '18 at 6:46

Since a singleton in $\mathbb{R}$ is a degenerate interval, defining$$ A_m = \{ k \in \mathbb{Z} \mid 1 \leqslant k \leqslant m,\ (k, m) = 1 \},\\ f_n(x) = \sum_{m = 1}^n \sum_{k \in A_m} \frac{k}{m} χ_{\{\frac{k}{m}\}}(x), $$ then $\{f_n\}$ is an increasing sequence of step functions and converges to $f$ pointwise.

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  • $\begingroup$ Can you tell me what is the geometrical approach behind approximating general measurable functions by a sequence of step functions? $\endgroup$ – Dbchatto67 May 6 '18 at 13:04
  • $\begingroup$ @DebabrataChattopadhyay. I don't know if this constitutes geometric intuition, but the idea is to first divide the domain of $f$ into intervals of certain length and then replace the value of $f$ by some value in each interval, thus resulting in a step function. By choosing finer and finer intervals, a sequence of step functions for approximation is derived. $\endgroup$ – Saad May 6 '18 at 14:00

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