Pointwise convergence of an increasing sequence of step functions to a measurable function.

Question

Give an example of a measurable non-negative function $f$ such that no incresing sequence of step functions which converges pointwise everywhere to $f$ exists.

I have tried to the best of my ability to find the example of such a measurable function but inspite of all my efforts I failed to find that example. Please help me in this regard. It will then really helpful for me.

Thank you very much.

Answer 1

If by step function you mean this definition :

$f$ is a step function if it can be written as $f = \sum_{k=1}^{n}\alpha_k\chi_{A_k}$ where $A_k\subseteq \mathbb{R}$ are intervals.

Note that $\{\alpha\}=[\alpha,\alpha]$ is an interval.

I think the indicator of $\mathbb{Q}\cap[0,1]$ wont work because if we take an enumeration of $\mathbb{Q}\cap[0,1] = \{q_1,q_2,...,q_n,...\}$ then the step functions defined by $$f_n(x)=\sum_{k=1}^{n}\chi_{\{q_k\}}(x)$$ have the desired properties.

I think the key point here is that the pointwise limit of step functions must be Borel measurable function.

So if we take a set $B$ which is Lebesgue measurable but not Borel measurable.

Then the indicator $\chi_B$ it cannot be a pointwise limit of step functions.

Edit: Also the indicator $\chi_{\mathbb{R}\smallsetminus\mathbb{Q}}$ works too since the pointwise limit of any sequence $0\leq f_n \leq \chi_{\mathbb{R}\smallsetminus \mathbb{Q}}$ of step functions must have countable non zero points.