Consider collections $ \mathcal{V} $ of functions from $ \mathbb{R} $ to $\mathbb{R} $ satisfying the following conditions:
(a) $ \mathcal{V} $ is a vector space
(b) $ \mathcal{V} $ contains the continuous functions
(c) If $ (f_{n})_{n} $ is an increasing sequence of nonnegative functions in $ \mathcal{V} $ and if $ \lim_{n \rightarrow \infty} f_{n}(t) $ exists and is finite for all $ t \in \mathbb{R} $, then $ \lim_{n\rightarrow \infty} f_{n}(t) \in \mathcal{V} $ .
Show that the collection $ \mathcal{V}_{0} $ consisting of the Borel-measurable functions is the smallest such collection of functions. (Hint: define $\mathcal{A}= \{ A \subseteq \mathbb{R}: \chi_{a} \in \mathcal{V} \}$. Show that $ \mathcal{A} $ contains the interval $ (-\infty,a) $, and then contains the Borel sets).
$\chi_{A} :$ characteristic function of $A$.
$\chi_{A}(x)=1$ if $x \in A $ , $\chi_{A}(x)=0$ if $x \notin A, $
I need to find or prove the existence of a sequence of positive continuous functions, increasing to converge to $ \chi_ {A} $, then I could use the basic theorem of measure theory to put all the functions borel measurable in the set. Any help is appreciated, I try with step functions but dont result.