Confusion about the index of a sequence of simple functions, that approximates a measb. function.

Question

It is known that a measurable function $f$ can always be approximated by a sequence of simple functions $f_n\uparrow f$ (pointwise). We can represent a simple function as $g=\sum_{i=1}^mx_i\bf{1}_{A_j}$ where $\bf{1}_{A_j}$ the indicator function. My question: if we have $g_n\uparrow f$ do we mean that $n=m$ so that we have,$f=\sum_{i=1}^{\infty}x_i\bf{1}_{A_j}$? The latter is not a simple function annymore since it doesn't take finite values. Or doesn't it have to be the case that $m=n$?

Answer 1

Not at all. Take a strictely increasing function on $\mathbb R$. Then, $$f(x)=\lim_{n\to \infty }\sum_{k=-n^2}^{n^2}f(k/n)\boldsymbol 1_{[\frac{k}{n},\frac{k+1}{n}]}(x),$$ but obviously, $f$ won't be of this form.