Let $G$ be a class of $\it{indicator}$ functions where $g\in G$ implies that $g : X \rightarrow \{0,1\}$, where the domain $X$ is a compact subset of $\mathbf{R}^n$. For example:
$$ g(x) = 0 \text{ if } w_1x_1+\cdots+w_dx_d < \eta $$ $$ g(x) = 1 \text{ otherwise } $$
Note: I am not interested in this particular class. This is just an example.
Is there a theorem that would say something like below?
Theorem: Let $G$ be an arbitrary class of $\it{indicator}$ functions with properties IMPORTANT TEXT MISSING. Then for any $f : X \rightarrow \mathbf{R}$ which is NICELY BEHAVING IN SOME SENSE and any $\epsilon$ a linear combination $h=w_1g_1+\cdots+w_kg_k$ (of elements in $G$) can be found such that $\vert h(x)-f(x)\vert<\epsilon$ for all $x\in X$.
Any hint highly appreciated!
EDIT: A possible answer might be found here but I am not sure. It all depends whether my understanding of the discussed theorem is correct.
For sake of an answer, let's assume by nicely behaved you mean "piecewise continuous and bounded functions on a compact subset of $R^n$."
The metric you've stated an interest in is the uniform convergence metric.
Arbitrary indicator functions can approximate piece wise continuous functions uniformly. For example, if you are interested in an $\epsilon$ approximation, create a partition $r_i$ of the real line with norm less than $\epsilon.$ Then let your indicator sets $A$ be $A_i = \{ x | r_{i-1} <= f(x) < r_i\}$.
But, you seem to be interested in a space that's more restrictive than "half open / closed subsets of $R^n$" (which is what is required for the above to work.)
The space $G$ has to be able to generate arbitrarily small sets, that don't overlap (so half open intervals on the dyadics are a good example). For continuous functions that is a sufficient characterization.
For piece wise continuous functions, your space $G$ has to be able to "exactly" model the discontinuities. So, for example, in $R^2$ if you have discontinuities on the line $y=x$, you cannot piecewise approximate it if the space of $G$ indicator functions is made up of half open rectangles.