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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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    how is it related to integral of differential forms coming from differential geometry ? – 2017-01-06
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    As already pointed out, $\xi(\text{d}x)$ is a notation to emphasize with respect to which measure the integral is taken and basically this has nothing to do with differential forms. – 2017-01-06
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    For the relation of differential forms with measures see for example https://en.wikipedia.org/wiki/Differential_form#Relation_with_measures or these questions: http://math.stackexchange.com/questions/57606/integral-of-differential-form-and-integral-of-measure http://math.stackexchange.com/questions/49641/integration-of-forms-and-integration-on-a-measure-space?noredirect=1&lq=1 – 2017-01-06