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In measure theoretic probability I often see these two notations for the expectation of a random variable expressed as an abstract integral.

$$ \int_\Omega X(\omega) \mathbb{dP(\omega)} = \int_\Omega X(\omega) \mathbb{P(d\omega)} $$

Is each case trying to express a different view of the abstract integral? What is an intuitive way to interpret $\mathbb{dP(\omega))}$ and $\mathbb{P(d\omega)}$?

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Given a probability space $(\Omega, \Sigma, \mathbb P)$ and a real-valued measurable function $X$ defined on $\Omega$, the symbols $$\int_\Omega X(\omega) \mathbb{dP(\omega)},\ \int_\Omega X(\omega) \mathbb{P(d\omega)}$$ are simply interchangeable notation and are defined to represent integration of the function $X$ with respect to the measure $\mathbb P$, also denoted as the symbol $\mathbb E[X]$ and called the expectation of $X$. "Intuition" cannot be gleaned from these symbols without understanding their underlying definition.

To put it bluntly: there is no meaning of the symbols $\mathbb{dP(\omega))}$ and $\mathbb{P(d\omega)}$ that is separate from them being part of the well-defined integral symbols above.

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  • $\begingroup$ Thank you. I understand the technical definitions of integration of $X$ with respect to $\mathbb{P}$. When $X$ has a density $f$, the expectation $\int_{x \in Im(X)} x f(x)dx$ is commonly explained as a weighted sum of the values $x$ that $X$ can take, where the value $f(x)dx$ is the probability that $X$ is in the neighborhood of $x$, or approximately $\mathbb{P}(X \in [x, x+dx])$ $\endgroup$
    – jII
    Apr 3, 2015 at 1:23
  • $\begingroup$ @jesterII, You are welcome. The limit of weighted sums of $X$-values is also used in a measure-theoretic treatment (the sums are integrals of so-called simple functions), where the weights are the probability of $X$ taking the associated value in an associated so-called Borel set. However, in my opinion, I will be doing you a great disservice, were I to imbue the symbols you asked about with any more meaning other than that which follows by definition (e.g. en.wikipedia.org/wiki/Lebesgue_integration#Integration) $\endgroup$
    – ki3i
    Apr 3, 2015 at 13:42

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