Lebesgue–Stieltjes measure

Remarks

Existence comes from the Carathéodory extension theorem applied to the premeasure $(a, b] \mapsto F(b) - F(a)$ on half-open intervals; right-continuity is what makes it countably additive. Uniqueness holds because $\R$ is a countable union of intervals $(-n, n]$ of finite measure. Many authors also call the completion of $\mu_F$ the Lebesgue–Stieltjes measure. Example: $F(x) = x$ gives Lebesgue measure on the Borel sets. Example: for $F = \mathbf{1}_{[0, \infty)}$, which is right-continuous, $\mu_F$ is the Dirac measure at $0$, and $\mu_F(\set{a}) = F(a) - \lim_{x \to a^-} F(x)$ in general. Non-example: $F(x) = -x$ is not non-decreasing, and the formula would give $(0, 1]$ the negative value $-1$. Two functions $F$ and $G$ give the same measure if and only if $F - G$ is constant. The notation $dF$ is common, as in $\int g \, dF$.