This page uses the nondecreasing, right-continuous, $(a,b]$-interval convention

This page fixes the convention

${\mu }_F((a,b])=F(b)-F(a)$

for a nondecreasing, right-continuous function $F:\mathbb{R}\to \mathbb{R}$. That pairing of monotonicity, interval shape, and continuity direction is the one used in Folland and Hunter, and it is the one for which the shrinking intervals $(a,a+1\mathrm{/}n]$ force right continuity through continuity from above.

Another standard convention in the literature uses the left-continuous data $[a,b)$ instead. The two choices are both legitimate, but they are not the same construction for a fixed function. This page adopts the former convention once, records the latter here only as a warning, and thereafter uses the left-limit notation $F(a^{-})$ of The left and right limits of $f$ at $c$, as limits of the restrictions of $f$ to $A\cap (-\mathrm{\infty },c)$ and $A\cap (c,\mathrm{\infty })$ only in derived interval formulas. Throughout this page, the order hypothesis on $F$ is the weak one $F(x)\le F(y)$ for $x\le y$.