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I was trying to make sense of the notation $P(X \in dx),$ where $X$ is a continuous random variable. Some also write this one as $P(X \in [x, x+dx])$ to represent the probability that the random variable $X$ takes on values in the interval $[x, x+dx].$

I have seen similar notation a lot in my readings such as the ff:

$$

P(Z_v \in dy) = \frac{1}{\Gamma(v)}e^{-y}y^{v-1} dy \quad\quad \text{and}\quad\quad

P\left( \int_0^{\tau} e^{\sigma B_s - p\sigma^2 s/2} ds \in du,\,\, B_{\tau} \in dy\right).

$$

Please help me understand these notations, or better yet please suggest me a (reference) book that rigorously explains these notations?

Thanks in advance.