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  "body": "\n## Definition\n\nLet $F \\colon \\R \\to \\R$ be a [[non-decreasing|0A7KV2WB7A]] function that is right-continuous at every point, that is, a [[Stieltjes function|WPR0NVC9PC]].\nThe *Lebesgue–Stieltjes measure* of $F$ is the unique [[measure|PR11BZQQPG]] $\\mu_F$ on the [[Borel sigma-algebra|0WQXY32Q8D]] of $\\R$ such that\n$$\\mu_F((a, b]) = F(b) - F(a) \\quad \\text{for all } a \u003c b.$$\n\n## Remarks\n\nExistence 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.\nUniqueness holds because $\\R$ is a countable union of intervals $(-n, n]$ of finite measure.\nMany authors also call the completion of $\\mu_F$ the Lebesgue–Stieltjes measure.\nExample: $F(x) = x$ gives Lebesgue measure on the Borel sets.\nExample: 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.\nNon-example: $F(x) = -x$ is not non-decreasing, and the formula would give $(0, 1]$ the negative value $-1$.\nTwo functions $F$ and $G$ give the same measure if and only if $F - G$ is constant.\nThe notation $dF$ is common, as in $\\int g \\, dF$.\n",
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