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Quick question that has been bugging me for a while: I have come across both the notation $dP(x)$ as well as $P(dx)$ when looking at the $\mathbb E[X]$ where $X$ is a continuous real random variable. Am I missing an important concept as to why both are used, rather than just one? Or are these notations simply equivalent?

SABOY
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    Neither of them means anything in their own except that the integral is with respect to the measure $P.$ In fact, you can drop the "$dx$" symbols everywhere and simply write $\mathbf{E}(X) = \int\limits_\Omega X dP.$ – William M. Jan 29 '19 at 20:27
  • They come handy when you have a function of two variables, say $x \in \Omega$ and $t \in \mathrm{T}$ and the measure $P$ is on $\Omega.$ But even then the fact that $P$ is defined on $\Omega$ should suffice... – William M. Jan 29 '19 at 20:28

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