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Reading advanced probability theory book I've come across Lebesgue-Stieltjes measure. Could someone explain what is the difference between it and "standard" Lebesgue measure on $\mathbb{R}$? Thank you.

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  • $\begingroup$ Lebesgue integral: $\int f(x)\;dx$. Lebesgue-Stieltjes integral: $\int f(x)\;dg(x)$. $\endgroup$
    – GEdgar
    Jan 8, 2019 at 1:09

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It can be proven that for any increasing right-continuous function $F:\mathbb{R}\rightarrow \mathbb{R}$ there exists a unique measure $\mu_F$ on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ satisfying the property: $$ \mu_F((a,b])=F(b)-F(a) $$ for every interval (a,b] with $a<b$.

$\mu_F$ is called the Lebesgue-Stieltjes measure belonging to F. The Lebesgue measure is simply the Lebesgue-Stieltjes measure belonging to the identity function.

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