Cumulative Riemann sums, distribution functions, and a universal inequality

We study discrete expressions of the form $$T_n(g)=\sum _{i=1}^n{a}_{i}g(S_i),S_i=\sum _{j=1}^i{a}_j,$$ where $a_i>0$ and ${\sum }_{i=1}^n{a}_i=1$. If $g:[0,1]\to \mathbb{R}$ is a decreasing integrable function, we have $$\sum _{i=1}^n{a}_{i}g(S_i)\le {\int }_0^{1}g(x)dx,$$ from which classical inequalities can be obtained, for instance for the choice $g(x)=1-x^k$. Although elementary, this inequality admits a natural interpretation in terms of Riemann sums, Abel summation, and the probability integral transform. The aim of this paper is to present a unified perspective emphasizing that the discrete inequality is a consequence of a distribution-free continuous identity. Beyond the specific example, we establish a general discrete theorem for monotone functions and discuss connections with majorization theory and Karamata’s inequality.