Let $X$ be a compactly generated topological space, and put $\text{Top}(X, \mathbb{R})$ for the algebra of continuous real valued functions on $X$.
An integral (induced by a measure) is a continuous linear map $\int : \text{Top}(X, \mathbb{R}) \rightarrow \mathbb{R}$.
Does this axiomatize integrals? Is there a reference for this perspective on measure theory? Thanks very much!