I am reading the paper.
I have some about distributions on $l$-spaces. On page 7, Section 1.7. Let $X$ be an $l$-space. Locally constant complex-valued functions on $X$ with compact support are called Schwartz functions on $X$. The space of Schwartz functions on $X$ is denoted by $S(X)$. The space of distributions on $X$ is denoted by $S^*(X)$ (distributions on $X$ are linear functionals on $S(X)$). Let $f \in S(X)$, $T \in S^*(X)$. Then the value of $T$ at $f$ is denoted by $\langle T, f \rangle$ or $\int_X f(x) dT(x)$.
My question is: why we denote the value of $T$ at $f$ by an integral $\int_X f(x) dT(x)$? Thank you very much.