Actually, I have ever posted about this theorem.
However, the reason why I asked this is I want to make sure about this theorem.
In my textbook referred, (I copied the theorem exactly same as the textbook because my poor English can change the original theorem),
$(4.13)$ Theorem
- Every function $f$ can be written as the limit of a sequence $\{f_k\}$ of simple functions.
- If $f\ge0$, the sequence can be chosen to increase to $f$, that is, chosen such that $f_k \le f_{k+1}$ for every $k$.
- If the function $f$ in either (i) or (ii) is measurable, then the $f_k$ can be chosen to be measurable.
However, my professor wrote on the board,
$f \geqslant 0$: mble $\Rightarrow$ $\exists$ simple function $f_k \nearrow f$.
I think $f$ need not be nonnegative. Should it be nonnegative like professor wrote?
(Add. Simple function's definition)
The characteristic function, $\chi_E(\mathbf{x})$, of a set $E\in\Bbb{R}^n$ is defined by $$\chi_E(\mathbf{x})=\cases{1\quad \text{if } \mathbf{x}\in E \\ 0\quad\text{if } \mathbf{x} \notin E}.$$
Clearly, $\chi_E$ is measurable if and only if $E$ is measurable. $\chi_E$ is an example of what is called a simple function: a simple function is one which assumes only a finite number of values, all of which are finite. If $f$ is a simple function taking (distinct) values $a_1, \cdots, a_N$ on (disjoint) sets $E_1, \cdots, E_N$, then $$f(\mathbf{x})=\sum^N_{k=1} a_k\chi_{E_k}(\mathbf{x})$$
Best Answer
If your definition of simple function does not allow infinite values, then you clearly can have a function with a single point mapped to $-\infty$ that is never bounded below by any simple function.