Left-continuous cumulative function of an atomic measure ($F(x)=\mu ((-\infty ,x))$)

For positive summable weights $w_j$ at points $q_j$, the strict cumulative function $F(x)={\sum }_{q_j<x}{w}_j$ is nondecreasing and left-continuous. A finite-head/tail argument proves left-continuity, and its right jump at an atom is the mass there. This strict convention differs at atoms from the usual right-continuous cumulative distribution function.