$\mathbf{Definition \ 1:}$ The outer Lebesgue measure $m^*(S)$ of a set $S\subset \mathbb{R}^n$ is
$$m^*(S):=\inf_{S \subset \cup_{i=1}^\infty B_i} \sum_{i=1}^\infty |B_i|$$
Where $B_i$ are boxes and $|B_i|$ denotes the usual elementary measure of the box $B_i$
$\mathbf{Definition \ 2:}$ A set $S\subset \mathbb{R}^n$ is Lebesgue measurable, if for every $\epsilon>0$, there exists an open set $U\subset \mathbb{R}^n$ containing $S$, such that $m^*(U\backslash S) < \epsilon$
$\mathbf{Problem:}$ Show a function $f:\mathbb{R}^n\rightarrow [0,\infty]$ is simple, if and only if it is Lebesgue measurable and takes on only finitely many values.
I am not sure how to go about the problem, and honestly have no clue where to start.
A complex valued simple function is a mapping $f:\mathbb{R}^n \rightarrow \mathbb{C}$, that is a finite linear combination of indicator functions of lebesgue measurable sets in $\mathbb{R}^n$. I.e. $f=\sum_{i=1}^n a_i 1_{S_i}$ where $a_i\in \mathbb{C}$ and $S_i$ are Lebesgue measurable in $\mathbb{R}^n$.
I apologize for the lack of work, and would appreciate any help. Thanks in advance!
Any sum of the type $\sum\limits_{i=1}^{n} a_i I_{S_i}$ with $S_i$'s measurable can be expressed in the form $\sum\limits_{i=1}^{m} b_i I_{T_i}$ where $T_i$'s are disjoint and $T_i$'s measurable. If $f=\sum\limits_{i=1}^{m} b_i I_{T_i}$ then $f$ takes only the values $b_1,...b_m$ and $0$ and it is Lebesgue measurable because inverse image of any Borel set $B$ is a union of those $T_i$'s for which $b_i \in B$. Conversely, if $f$ has these properties and $b_1,b_2,...,b_n$ are the distinct values of $f$ then $f=\sum\limits_{i=1}^{m} b_i I_{T_i}$ where $T_i=f^{-1}(\{b_i\})$.