More specifically, as an educational tool I want to prepare a slideshow showing (in 2-D) the graph of $F_X$ transforming into the graph of $F_G$. (I think it can be done in 4 steps (e.g., graphs on $R$, then $X$, then $G$, and back up to $R$) and am looking for rigor behind the visuals.)
The thought process:
A random variable, X, is defined as a particular type of function from a probability space to the reals, its image the support of its cdf, FX. Thus, “X” denotes a function and, in a supporting role, a set. Given random variable, G=g(X), transform the graph of FX to that of FG in R2 as follows:
1.{(r,FX(r)): rϵR} to {(x,FX(x)): xϵX}: We simply highlight the sub-graph for X.
2.{(x,FX(x)): xϵX} to {(g(x),FX(x)): xϵX}: For each (x,FX(x)) plot (g(x),FX(x)).
3.{(g(x),FX(x)): xϵX} to {(g,FG(g)): gϵG}: We arrive at the sub-graph for G by “sorting” the (g(x),FX(x)) in place vertically by the size of g(x): the resulting 2nd coordinate in (g,FG(g)) being the “sum” over all the F(x)-F(x-) for those (g(x),FX(x)) for which g(x)≤g .
4.{(g,FG(g)): gϵG} to {(r,FG(r)): rϵR}: As G is the support, if rϵR and not in G then FG(r)=0 or FG(r)=FG(g_r) where g_r is the largest value in G less than r.
At issue in particular is the rigor/validity of Step 3.