Properties of Simple Functions

Question

Suppose $f$ is a non-negative measurable function. i.e $f \in L^+$.

1. Is it true that f is the decreasing limit of a sequence of simple functions?

I'm suspecting that it's false, since it's we don't know if $f$ is bounded.

But I couldn't find any counter examples.

2. is the set of real valued simple functions closed under taking the max and min of two functions?

I'm also suspecting that it's false, because the limit of simple functions may not be simple. But I'm not sure about this one.

Thanks in advance!

Answer 1

1. Real-valued simple functions are bounded, because they are the measurable functions with finite range. Therefore no unbounded positive function can be smaller than a simple function. If you allow simple functions taking values in $[0,+\infty]$, then yes you can: with the convention of convenience $\frac10=+\infty$, approximate from below $\frac1f$ with a sequence of simple functions $0\le\varphi_n\nearrow \frac1f$, then consider the sequence $\phi_n=\frac1{\varphi_n}$.

2. Yes, because the maximum and minimum of two measurable functions with finite range are measurable functions with finite range (in fact, their ranges are subsets of the union of the ranges).