I am self-studying stochastic processes. The book I am using states the following:
Let $x<y$ and $y\geq0$. Then $$Pr(W_t\leq x,M_t^W\geq y)=Pr(W_t \geq 2y-x,M_t^W\geq y)$$ hence $$Pr(W_t \in dx,M_t^W \in dy) = Pr(2y-W_t \in dx,M_t^W \in dy) $$ It follows that $$\mathbb{E}\left[exp\left(\nu W_t-\frac{1}{2}\nu^2 t\right)\; 1_{\{W_t\leq x,M_t^W\geq y\}}\right] = \mathbb{E}\left[exp\left(\nu(2y-W_t)-\frac{1}{2}\nu^2 t\right)1_{\{W_t\geq 2y-x,M_t^W\geq y\}}\right]$$
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.
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 $$P(X\in dx, Y\in dy)=\mu({dx,dy})=f_{X,Y}(x,y)\,dx\,dy.$$