It seems that the definition of convex hull is on a form $\sum a_nx_n$ in which coefficients sum up to 1. It implicitly implies that the combination is countable and discrete. I am interested in more abstract expression: $\int x da$ where $\int da =1$. Let me call it generalized convex hull. Are there some known results on the generalized convex hull? I believe it contains the convex hull but still a subset of closed convex hull. However, I cannot find a good example in $R^n$ to understand the concept. I think the generalized convex hull of open unit disk $\{(x,y):x^2+y^2<1\}$ is still the open unit disk? Am I right? If I am right, then we have Convex Hull$=$Generalized Convex Hull$\subset$Closure of Convex Hull in this case.
Another example I find interesting is suppose $A =\{\mathbf{1}_{[0,pt]}(x):pt\in [0,1],x\in [0,1]\} $. Then the convex combination of $A$ is just step functions and they are all discontinuous. However, generalized convex hull of $A$ contains functions which are continuous in $x\in(0,1)$. Am I right? If I am correct, then what is the closure of convex hull of $A$? It seems now that closure is just the generalized convex hull of $A$. Hence we have Convex Hull$\subset$Generalized Convex Hull$=$Closure of Convex Hull. As for topology in this example, we can use that induced by supremum norm on $[0,1]$.
When is that generalized convex hull and convex hull are equivalent and when is that closure of convex hull and generalized convex hull are equivalent?