I'm having trouble judging whether this statement is correct:
For an arbitrary bounded measurable function $f$ defined on $[0,1]$, $\exists{}\ $a sequence of step functions $\{\phi_n\}$, such that $\{\phi_n\}$ converges to $f$ pointwisely a.e. on $[0,1]$.
By the Simple Approximation Theorem, this is true if we are allowed to use simple functions. But I am curious whether this still holds when we restrict ourselves to step functions only.
I have a feeling that this may not be true because for a measurable function, its domain may be too "broken up" to be fitted by step functions. But I don't know how to find a counter-example...
So can anybody help me find a counterexample or confirm that this is correct?
Thank you very much!
Edit: By a step function I mean a (finite) linear combination of indicator functions for intervals.