Abstract Integration in Elementary Probability Theory 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 $\mathbb{P(d\omega)}$?

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.
