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I am self-studying stochastic processes. The book I am using states the following:

Let $x

I am confused by this notation and then by the equality between the two expectations. I have never encountered this notation of $P(X\in dx)$ but I presume it means $P(X \in[x,x+\delta])$ for some small $\delta$, in which case the first lines make sense. Is that right?

Can I think of $P(X\in dx)$ as the density of X if such exists?

Are there are references where I can learn more about this notation?

But then I am still confused by the equality of expectations. Maybe it is trivial but I would like to derive it from the definition of all the objects. It looks as some sort of substitution, but I am not sure how it works in this setting, which is more abstract than usual statistics. For example, in calculus approach to substitution of random variables, in integrals we would have Jacobian of such substitution. But I don't know what would be Jacobian here.

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In general, $P(X\in dx, Y\in dy)$ is common jargon in probability theory for $\mu({dx,dy})$, where $\mu$ is the joint distribution of the random vector $(X,Y)$. This notation is most commonly adopted when taking expected values. For instance, $$\int_{\mathcal{X\times\mathcal{Y}}} f(x,y) P(X\in dx, Y\in dy)=E(f(X,Y)).$$

Indeed, if $\mu$ is absolutely continuous w.r.t. Lebesgue measure, say with density $f_{X,Y}(x,y)$ then $$P(X\in dx, Y\in dy)=\mu({dx,dy})=f_{X,Y}(x,y)\,dx\,dy.$$

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    Thanks for taking time to reply! That makes sense regarding the notation from the measure-theoretic point of view, But I am still trying to understand how one uses this notation in applications (like in the book I am using). For example, suppose that X and Y have a joint density $p_{X,Y}(x,y)$. Is then the case that $P(X\in dx,Y \in dy)=p(x,y)dx dy?$. – 2017-01-16
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    Precisely, since the two expressions are linked via $\mu$. I have edited this in. – 2017-01-16