While reading a paper related to functional of brownian motion I came across the following notation $1(B_t \in dx)$, where $1(A)$ is the indicator function of the set A, and $B_t$ is a standard brownian motion.
...Trying to make sense of the expectation of this random variable I write informally: $$E[1(B_t \in dx)]=\int_R 1(X_t \in dx)f_{X_t}(x)dx=\int_{a}^{a+dx}f_{X_t}(x)dx=f_{X_t}(a)dx$$ with $f_{X_t}(x)dx$ the density function of std brownian motion at time t. I have no doubt in my mind that the last 2 equalities makes little sense, but I can't really figure what it should be instead.
Appreciate any pointers, thanks.