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In the context of measure theory, given a probability measure $\xi : \mathcal{B}(X) \rightarrow [0,1]$ and a (smooth) function $v:X\rightarrow \mathbb{R}$ where $X\subset \mathbb{R}^n$, we encounter the notation

$$ \int_X v(x)\, \xi(dx) $$

What is behind the notation $\xi(dx)$ formally ? Do probability measures act on differential forms ? If so, what is this action formally ?

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If $v$ is a measurable function, the integral in your question is just a notation for the Lebesgue integral of the function $v$ with respect to the measure $\xi$. Other standard notations are $$ \int_{X} v(x) \, \xi(dx):= \int_{X} v(x) \, d\xi(x) := \int_{X} v \, d\xi. $$

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