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I sometimes read the notation $$P[X \in d\omega].$$ I don't know what it means, how to pick the elements, the random variable is mapped to, to a "set" (which is none), or more precisely a variable with respect to we are integrating.

Hope you can help me.

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It's a notation for the pushforward measure of $X$:

$$\mu(B) := \mathbb{P}[X \in B] = \mathbb{P} (\{w \in \Omega; X(w)\in B\})$$


$$\int f(X) \, \mathbb{P}(dw)=\int f \, \mathbb{P}[X \in dw] = \int f \, \mu(dw)$$


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